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5

Patterns and Collective Behavior in Granular Media: Theoretical Concepts

Igor S. Aranson∗

Materials Science Division, Argonne National Laboratory, 9700 S. Cass Av, Argonne, IL 60439

Lev S. Tsimring†

Institute for Nonlinear Science, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093

(Dated: February 2, 2008)

Granular materials are ubiquitous in our daily lives. While they have been a subject of intensiveengineering research for centuries, in the last decade granular matter attracted significant attentionof physicists. Yet despite a major efforts by many groups, the theoretical description of granularsystems remains largely a plethora of different, often contradicting concepts and approaches.Authors give an overview of various theoretical models emerged in the physics of granular matter,with the focus on the onset of collective behavior and pattern formation. Their aim is two-fold:to identify general principles common for granular systems and other complex non-equilibriumsystems, and to elucidate important distinctions between collective behavior in granular andcontinuum pattern-forming systems.

Contents

I. Introduction 1A. Preliminary remarks 1B. Fundamental microscopic interactions 2

II. Overview of dynamic behavior in granular matter3

A. Pattern formation in vibrated layers 3B. Gravity-driven granular flows 4C. Flows in rotating cylinders 4D. Grains with complex interactions 4

III. Main theoretical concepts 5A. Kinetic theory and hydrodynamics 5B. Phenomenological models 7C. Molecular dynamics simulations 7

IV. Patterns in sub-monolayers. Clustering, Coarsening and Phase Transitions8

A. Clustering in Freely Cooling Gases 8B. Patterns in Driven Granular Gases 9C. Coarsening of clusters 10

V. Surface waves and patterns in vibrated multilayers of granular materials11

A. Chladni patterns and heaping 11B. Standing wave patterns 11C. Simulations of vibrated granular layers 12D. Continuum theories 13

VI. Patterns in gravity-driven dense granular flows 16A. Avalanches in thin granular layers 16

1. Partially fluidized flows 162. Two-phase flow approach of granular avalanches 173. Avalanche shape 18

B. Statistics of avalanches and sandpile model 18C. Instabilities in granular chute flows 19D. Pattern-forming instabilities in rotating cylinders 20

VII. Models of granular segregation 21

∗Electronic address: aronson@msd.anl.gov†Electronic address: ltsimring@ucsd.edu

A. Granular stratification 21B. Axial segregation in rotating drums 22C. Other examples of granular segregation 25

VIII. Granular materials with complex interactions 26A. Patterns in solid-fluid mixtures 26B. Vortices in vibrated rods 28C. Electrostatically driven granular media 30

1. Coarsening of clusters 302. Dynamics of patterns in a fluid-filled cell 31

D. Magnetic particles 32

IX. Overview and Perspectives 33

Acknowledgments 35

References 35

I. INTRODUCTION

A. Preliminary remarks

Granular materials are ubiquitous in our daily livesand basic to many industries. Yet understandingtheir dynamic behavior remains a major challenge inphysics, see for review de Gennes (1999); Duran (1999);Gollub and Langer (1999); Jaeger et al. (1996); Kadanoff(1999); Nedderman (1992); Ottino and Khakhar (2000);Rajchenbach (2000); Ristow (1999). Granular materialsare collections of discrete macroscopic solid grains withsizes large enough that Brownian motion is irrelevant (en-ergy of 1 mm grain moving with typical velocity of 1cm/sec exceeds the thermal energy at least by 10 ordersof magnitude). Since thermodynamic fluctuations do notplay a role, for granular systems to remain active theyhave to gain energy either from shear or vibration and arethus far from equilibrium. External volume forces (grav-ity, electric and magnetic fields) and flows of interstitialfluids such as water or air may also be used to activate thegrains. When subjected to a large enough driving force, agranular system may exhibit a transition from a granular

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solid to a liquid and various ordered patterns of grainsmay develop. Understanding fundamentals of granularmaterials draws upon and gives insights into many fieldsat the frontier of modern physics: plasticity of solids,fracture and friction; complex systems from equilibriumsuch as colloids, foams, suspensions, and biological self-assembled systems. Moreover, particulate flows are cen-tral to a large number of industries including the chemi-cal, pharmaceutical, food, metallurgical, agricultural andconstruction industries. Beyond these industrial applica-tions, particle laden-flows are widespread in nature, forexample dune migration, erosion/deposition processes,landslides, underwater gravity currents and coastal ge-omorphology, etc.

From a theoretical point of view, it is sometimes use-ful to employ an analogy between granular matter andordinary condensed matter and to regard the grainsas the equivalent of (classical) atoms. However, thisanalogy is far from complete, since the dissipative na-ture of grain interactions is the source of many differ-ences between the two “kinds” of matter. In partic-ular, dissipation is responsible for the fact that moststates of granular matter are metastable. The typicalmacroscopic size of the grains renders thermal fluctua-tions negligible and most standard thermodynamic con-cepts inapplicable. Whereas the behavior of dilute gran-ular systems (rapid granular gases) can often be ex-plained using the framework of kinetic theory (see e.g.Brilliantov and Poschel (2004)), the quantitative theoryof dense granular assemblies is far less developed.

In recent years several comprehensive reviewsand monographs have appeared on the subjectof granular physics, see (Aradian et al., 2002;Brilliantov and Poschel, 2004; Duran, 1999; Jaeger et al.,1996; Kudrolli, 2004; Ottino and Khakhar, 2000;Rajchenbach, 2000; Ristow, 2001). Yet in most of themthe focus has been on actual phenomena and experimentsrather than on theoretical concepts and approaches tothe problems of granular physics. Furthermore, thescope of granular physics has become so broad thatwe chose to limit ourselves with reviewing the recentprogress in a subfield of granular pattern formation

leaving out many interesting and actively developingsubjects. We loosely define pattern formation as adynamical process leading to the spontaneous emergenceof nontrivial spatially non-uniform structure which isweakly dependent on initial and boundary conditions.According to our working definition, we include in thescope of the review the patterns in thin layers of vibratedgrains (Sec. IV,V), patterns in gravity-driven flows (Sec.VI), granular stratification and banding (Sec. VII),as well as a multitude of patterns found in granularassemblies with complex interactions (Sec. VIII). Beforedelving into details of theoretical modelling of thesepattern-forming systems, we present a brief overview ofthe relevant experimental findings and main theoreticalconcepts (Sec. II and III).

B. Fundamental microscopic interactions

Probably the most fundamental microscopic propertyof granular materials is irreversible energy dissipation inthe course of interaction (collision) between the parti-cles. For the case of so-called dry granular materials,i.e. when the interaction with interstitial fluid such asair or water is negligible, the encounter between grainsresults in dissipation of energy while total mechanicalmomentum is conserved. In contrast to the interactionof particles in molecular gases, the collisions of macro-scopic grains is generally inelastic. There are severalwell-accepted models addressing the specifics of energydissipation in the course of collision, see for details e.g.Brilliantov and Poschel (2004). The simplest case corre-sponds to in-deformable (hard) frictionless particles withfixed restitution coefficient 0 < e < 1 characterizing thefraction of energy lost in the course of collision. Therelation between the velocities after the collisions (v′

1,2)and before the collision (v1,2) for two identical sphericalparticles is given by

v′1,2 = v1,2 ∓

1 + e

2[n12(v1 − v2)]n12. (1)

Here n12 is the unit vector pointed from the center ofparticle 1 to the center of particle 2 at the moment ofcollision. The case of e = 1 corresponds to the elasticcollisions (particles exchange their velocities) and e = 0characterizes fully inelastic collisions. For 0 < e < 1 thetotal energy loss is of the form

∆E = −1 − e2

4|n12(v1 − v2)|2.

Modelling collisions between particles by a fixed restitu-tion coefficient is very simple and intuitive, however thisapproximation can be questionable in certain cases. Forexample, approximation of granular media by a gas ofhard particles with fixed e often yields non-physical be-havior such as inelastic collapse (McNamara and Young,1996): divergence of the number of collisions in a finitetime, see Subsec. IV.A. In fact, the restitution coefficientis known to depend on the relative velocities of collidingparticles and approaches unity as |v1 − v2| → 0. Thisdependence is captured by the visco-elastic modelling ofparticle collision (see e.g. Ramirez et al. (1999)). Fornon-spherical grains the restitution coefficient may alsodepend on the point of contact (Goldsmith, 1964).

Tangential friction forces play an important role in thedynamics of granular matter, especially in dense systems.Friction forces are hysteretic and history dependent (thecontact between two grains can be either stuck due todry friction or sliding depending on the history of in-teraction). This strongly nonlinear behavior makes theanalysis of frictional granular materials extremely diffi-cult. In the majority of theoretical studies, the simplestCoulomb law is adopted: friction is independent on slid-ing velocity as long as tangential force exceeds the cer-tain threshold (Walton, 1993). However, the main prob-lem is represented by calculation of the static friction

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forces. It is well known that frictional contact forcesamong solid particles exhibit indeterminacy in case ofmultiple contacts per particle because there are less forcebalance constraints than stress components (see, e.g.McNamara et al. (2004); Unger et al. (2005)). To resolvethis indeterminacy in simulations, various approximatealgorithms have been proposed. In soft particle molecu-lar dynamics simulations the most widely used approachto calculating friction forces is the spring-dashpot model(Cundall and Strack, 1979; Schaffer et al., 1996). An-other approach is taken in the contact dynamics method.By assuming that all particles are rigid and treating allcontacting particles as performing instantaneous colli-sions (even those which are in fact in persistent contact),one can compute the contact forces generated duringthese collisions based on local force balance and impen-etrability of the particles constraints (see Brendel et al.

(2004); Moreau (1994)).Viscous drag forces due to interaction with intersti-

tial fluid often affect the dynamics of granular materials.Gas-driven particulate flows is an active research areain the engineering community, see e.g. Jackson (2000).Fluid-particle interactions are also involved in many geo-physical processes, e.g. dune formation (Bagnold, 1954).Whereas interaction of small individual particles withthe fluid is well-understood in terms of Stokes law, col-lective interaction and mechanical momentum transferfrom particles to fluid remains an open problem. Variousphenomenological constitutive equations are used in theengineering community to model fluid-particulate flows,see e.g. Duru et al. (2002).

Finally, small particles can acquire electric charge ofmagnetic moment. In this situation fascinating collec-tive behavior emerge due to competition between short-range collisions and long-range electromagnetic forces,see e.g. Aranson et al. (2000); Blair et al. (2003a);Sapozhnikov et al. (2003). Effects of complex inter-particle interactions on pattern formation in granular sys-tems will be discussed in Sec. VIII.

II. OVERVIEW OF DYNAMIC BEHAVIOR IN

GRANULAR MATTER

In this Section we give a brief overview of the mainexperiments illustrating the dynamical behavior of gran-ular media and the phenomena to be discussed in greaterdepth in the following Sections. We classify the exper-iments according to the way energy is injected into thesystem: vibration, gravity, or shear.

A. Pattern formation in vibrated layers

Quasi-two-dimensional sub-monolayers of grains sub-jected to vertical vibration exhibit a surprizing bimodalregime characterized by a dense cluster of closely packedalmost immobile grains surrounded by gas of agitated

FIG. 1 Top view of dense immobile cluster coexisting withdilute granular gas, from Olafsen and Urbach (1998).

FIG. 2 A typical clustering configuration in two dimensions,restitution coefficient 0.6, number of particles 40,000, fromGoldhirsch and Zanetti (1993).

particles, (Olafsen and Urbach, 1998), Fig. 1. This clus-tering transition occurs when the magnitude of vibra-tion is reduced (the system is “cooled down”) which isreminiscent to the clustering instability observed in non-driven (freely cooling) gas of inelastic particles discov-ered by Goldhirsch and Zanetti (1993), Fig. 2. Detailedconsideration of clustering phenomena in sub-monolayersystems is given in Sec. IV.

Multilayers of granular materials subject to vertical vi-bration exhibit spectacular pattern formation. In a typ-ical experimental realization a layer of granular materialabout 10-30 particle diameters thick is energized by pre-cise vertical vibration produced by an electromagneticshaker. Depending on experimental conditions, plethoraof patterns can be observed, from stripes and squares tohexagons and interfaces, see Fig. 3. While the first ob-servations of patterns in vibrated layers were made morethan two centuries ago by Chladni (1787) and Faraday(1831), the current interest in these problems was ini-tiated by Douady et al. (1989); Fauve et al. (1989) andculminated in the discovery by Umbanhowar et al. (1996)of a remarkable localized object, oscillon, Fig. 4. De-tailed consideration of these observations and their mod-elling efforts is given in Sec. V.

In another set of experiments pattern formation wasstudied in a horizontally vibrated system, see e.g.Liffman et al. (1997); Ristow (1997); Tennakoon et al.

(1998). While there are certain common features, suchas sub-harmonic regimes and instabilities, horizontallyvibrated systems do not show richness of behavior typ-ical for the vertically vibrated systems, and nontrivialflow regimes are typically localized near the walls. Whenthe granular matter is polydisperse, vertical or horizontalshaking often leads to segregation. The most well-knownmanifestation of this segregation is the so-called “Brazilnut” effect when large particles float to the surface ofa granular layer under vertical shaking (Rosato et al.,1987). Horizontal shaking is also known to produce inter-esting segregation band patterns oriented orthogonally tothe direction of shaking (Mullin, 2000, 2002) (see Fig. 5).

FIG. 3 Representative patterns in vertically vibrated granu-lar layers for various values of frequency and amplitude of thevibration: stripes, squares, hexagons, spiral, interfaces, andlocalized oscillons, from Umbanhowar et al. (1996).

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FIG. 4 Localized oscillon in vertically vibrate granular layer,from (Umbanhowar et al., 1996).

FIG. 5 Sequence of snapshots of a layer of copper balls/poppyseeds mixture in a horizontally shaken cavity (frequency 12.5Hz, amplitude 2 mm) at times 5 min, 10 min, 15 min, 30 min,1 h, 6 h, from Mullin (2000).

B. Gravity-driven granular flows

Gravity-driven systems such as chute flows andsandpiles often exhibit nontrivial patterns and spatio-temporal structures. Possibly the most spectacular areavalanches observed in the layers of granular matter ifthe inclination exceeds the critical angle (static angleof repose). Avalanches were a subject of continued re-search for many decades, however only recently it wasestablished that the avalanche shape depends sensitivelyon the thickness of the layer and the inclination an-gle: triangular downhill avalanches in thin layers andballoon-shaped avalanches in thicker layers which ex-pand both uphill and downhill, see Fig. 6 and Daerr(2001); Daerr and Douady (1999). Gravity-driven gran-ular flows are prone to a variety of non-trivial sec-ondary instabilities in granular chute flow: fingering(Pouliquen et al., 1997), see Fig. 7, longitudinal vor-tices in rapid chute flows (Borzsonyi and Ecke, 2005;Forterre and Pouliquen, 2001), see Fig. 8, long modu-lation waves (Forterre and Pouliquen, 2003), and others.

Rich variety of patterns and instabilities has also beenfound in underwater flows of granular matter: transverseinstability of an avalanche fronts, fingering, pattern for-mation in the sediment behind the avalanche, etc. (seeDaerr et al. (2003); Malloggi et al. (2005a,b)). Whereascertain pattern forming mechanisms are specific to thewater-granulate interaction, one also finds striking simi-larities with the behavior of “dry” granular matter.

C. Flows in rotating cylinders

Energy is often supplied into a granular systemthrough the shear which is driven by the moving wallsof the container. One of the most commonly used ge-ometries for this class of systems is a horizontal cylin-der rotated around its axis, or rotating drum. Rotatingdrums partly filled with granular matter are often used inchemical engineering for mixing and separation of parti-cles. Flows in rotating drums recently became a subject

FIG. 6 Sequence of images illustrating evolution of avalanchesin thin layers on incline. Three images on left: triangularavalanche in thin layer, point b in Fig. 24. Three right im-ages: up-hill avalanche in thicker layer, point c in Fig. 24,from Daerr and Douady (1999)

FIG. 7 Fingering instability in chute flow. (a) Schematicsof the instability mechanism, the arrows represent trajectoryof coarse particles. Images taken from front (b) and bottom(c) illustrating accumulation of coarse particles between theadvancing fingers, from Pouliquen et al. (1997).

FIG. 8 Development of longitudinal vortices inthe rapid granular flow down rough incline, fromForterre and Pouliquen (2001).

of active research in the physics community. For not toohigh rotating rates the flow regime in the drum is sepa-rated into an almost solid-body rotation in the bulk ofthe drum and a localized fluidized layer near the freesurface (Fig. 9). Slowly rotating drums exhibit oscilla-tions related to the gradual increase of free surface angleto the static angle of repose and subsequent fast relax-ation to a lower dynamic repose angle via an avalanche.Transition to steady flow is observed for the higher ro-tation rate (Rajchenbach, 1990). Scaling of various flowparameters with the rotation speed (e.g. the width ofthe fluidized layer etc) and development of correlationsin “dry” and “wet” granular matter was recently studiedby Tegzes et al. (2002, 2003).

Rotating drums are typically used to study sizesegregation in binary mixtures of granular materials.Two types of size segregation can be distinguished:radial and axial. Radial segregation is a relativelyfast process and occurs after a few revolutions of thedrum. As a result of radial segregation larger par-ticles are expelled to the periphery and a core ofsmaller particles is formed in the bulk (Khakhar et al.,1997; Metcalfe et al., 1995; Metcalfe and Shattuck, 1998;Ottino and Khakhar, 2000), see Fig. 10.

Axial segregation, occurring in the long drums, hap-pens on a much longer time scale (hundreds of rev-olutions). As a result of axial segregation, bands ofsegregated materials are formed along the drum axis(Hill and Kakalios, 1994, 1995; Zik et al., 1994), see forillustration Fig. 11. The segregated bands exhibit slowcoarsening behavior. Even more surprisingly, under cer-tain conditions axial segregation patterns show oscilla-tory behavior and travelling waves (Choo et al., 1997;Fiodor and Ottino, 2003). Possible mechanisms leadingto axial segregation are discussed in Sec. VII.

D. Grains with complex interactions

Novel collective behaviors emerge when the interac-tions between the grains have additional features causedby shape anisotropy, interstitial fluid, magnetization or

FIG. 9 Schematics of flow structure in the cross-section ofrotating drum, from Khakhar et al. (1997)

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FIG. 10 Radial size segregation in a rotating drum, courtesyof Wolfgang Losert.

FIG. 11 Long rotating drum showing axial size segregation,from http://www.physics.utoronto.ca/nonlinear/

electrical charge, etc. In this situation short-range col-lisions, the hallmark of “traditional” granular systems,can be augmented by long-range forces.

Remarkable patterns including multiple rotating vor-tices of nearly vertical rods are observed in the system ofvibrated rods by Blair et al. (2003a), see Fig. 12. Therods jump on their ends slightly tilted and drift in thedirection of the tilt.

Mechanically (Blair and Kudrolli, 2003b) or electro-statically (Snezhko et al., 2005) driven magnetic grainsexhibit formation of long chains, isolated rings or inter-connecting networks, see Fig. 13. In this situation mag-netic dipole-dipole interaction augments hard-core colli-sions.

Ordered clusters and nontrivial dynamic states wereobserved by Thomas and Gollub (2004); Voth et al.

(2002) in a small system of particles vibrated in liq-uid (Fig. 14). It was shown that fluid-mediated in-teraction between particles in a vibrating cavity leadsto both long-range attraction and short-range repul-sion. A plethora of nontrivial patterns including rotat-ing vortices, pulsating rings, chains, hexagons etc wasobserved by Sapozhnikov et al. (2003a) in the systemof conducting particles in dc electric field immersed inpoor electrolyte (Fig. 15). The nontrivial competi-tion between electrostatic forces and self-induced electro-hydrodynamic flows determines the structure of emergingpattern.

Granular systems with complex interactions serve asa natural bridge to seemingly different systems suchas foams, dense colloids, dusty plasmas, ferrofluids andmany others.

III. MAIN THEORETICAL CONCEPTS

Physics of granular media is a diverse and eclecticfield incorporating many different concepts and ideas,from hydrodynamics to the theory of glasses. Conse-quently, many different theoretical approaches have beenproposed to address observed phenomena.

FIG. 12 Select patterns observed in the system of verticallyvibrated rods with the increase of vibration amplitude: a)nematic-like gas phase; b) moving domains of nearly verti-cal rods; c) multiple rotating vortices; d) single vortex, fromBlair et al. (2003a).

FIG. 13 Structures formed in submonolayer of magnetic mi-croparticles subjected to alternating magnetic field. Selectstructures such as rings, compact clusters, and chains areshown in the top panel. Changes in the pattern morphologywith the increase of magnetic field frequency are illustratedby the three bottom images, from Snezhko et al. (2005).

FIG. 14 Regular arrangements of particles near the bottomof a vibrated container filled with water when both attractionand repulsion are important. All images are taken taken atthe frequency f = 20 Hz and for different value of dimension-less acceleration or for different initial conditions: (a) & (b)Γ = 3; (c) & (d) Γ = 3.7 and Γ: (e) Γ = 3.9 and (f) Γ = 3.5,from Voth et al. (2002).

A. Kinetic theory and hydrodynamics

Kinetic theory deals with the equations for the proba-bility distributions functions describing the state of gran-ular gas. The corresponding equations, similar to Boltz-mann equations for rarefied gases, can be rigorously de-rived for the dilute gas of inelastically colliding parti-cles with fixed restitution coefficient, although certaingeneralizations are known, (Goldstein and Shapiro, 1995;Jenkins and Zhang, 2002). Kinetic theory is formulatedin terms of the Boltzmann-Enskog equation for the prob-ability distribution function f(v, r, t) to find the particleswith the velocity v at point r at time t. In the simplestcase of identical frictionless spherical particles of radiusd with fixed restitution coefficient e it assumes the fol-lowing form

(∂t + (v1 · ∇)) f((v1, r1, t) = I[f ] (2)

with the binary collision integral I[f ] in the form

I = d2

∫

dv2

∫

dn12Θ(−v12 · n12)|v12 · n12| ×

[χf(v1′′, r1, t)f(v2

′′, r1 − dn12, t)

−f(v1, r1, t)f(v2, r1 + dn12, t)] (3)

where χ = 1/e2, Θ is theta-function, and pre-collisionvelocities v1,2 and “inverse collision” velocities v′′1,2 arerelated as follows

v′′1,2 = v1,2 ∓

1 + e

2e[n12(v1 − v2)]n12 (4)

(cf. Eq. (1)). This equation is derived with the usual“molecular chaos” approximation which implies that allcorrelations between colliding particles are neglected.

FIG. 15 Representative patterns obtained for different valuesof applied field and concentration of ethanol in electrostati-cally driven granular system: static clusters (a) and honey-combs (b) and dynamic vortices (c) and pulsating rings (d),from Sapozhnikov et al. (2003a)

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One should keep in mind, however, that in dense granu-lar systems this approximation can be rather poor due toexcluded volume effects and inelasticity of collisions in-troducing velocity correlations among particles (see, forexample, Brilliantov and Poschel (2004)).

Hydrodynamic equations are obtained by truncatingthe hierarchy of moment equations obtained from theBoltzmann equation (2) via an appropriately modifiedChapman-Enskog procedure (see, e.g., (Brey et al., 1998;Garzo and Dufty, 1999; Jenkins and Richman, 1985)).As a result, a set of continuity equations for mass,momentum and fluctuation kinetic energy (or “granu-lar temperature”) is obtained. However, in contrast toconventional hydrodynamics, the applicability of gran-ular hydrodynamics is often questionable because typi-cally there is no separation of scale between microscopicand macroscopic motions 1, see e.g. Tan and Goldhirsch(1998).

The mass, momentum and energy conservation equa-tions in granular hydrodynamics have the form

Dν

Dt= −ν∇ · u, (5)

νDu

Dt= −∇ · σ + νg, (6)

νDT

Dt= −σ : γ −∇ · q − ε, (7)

where ν is the filling fraction (the density of granularmaterial normalized by the density of grains), u is thevelocity field, T = (〈uu〉 − 〈u〉2)/2 is the granular tem-perature, D/Dt = ∂t + (u · ∇) is the material derivative,g is the gravity acceleration, σαβ is the stress tensor, q isthe energy flux vector, γαβ = ∂αuβ + ∂βuα is the strainrate tensor, and ε is the energy dissipation rate. Eqs.(5)-(7) are structurally similar to the Navier-Stokes equa-tions for conventional fluids except for the last term inthe equation for granular temperature ε which accountsfor the energy loss due to inelastic collisions.

These three equations have to be supplemented by theconstitutive relations for the stress tensor σ, energy fluxq, and the energy dissipation rate ε. For dilute systems,a linear relations between stress σ and strain rate γ isobtained,

σαβ = [p+ (µ− λ)Trγ]δαβ − µγαβ , (8)

q = −κ∇T. (9)

In the kinetic theory of two-dimensional gas of slightlyinelastic hard disks by Jenkins and Richman (1985),these equations are closed with the following equation

1 Except the case of almost elastic particles with the restitutioncoefficient e→ 1

of state

p =4νT

πd2[1 + (1 + e)G(ν)], (10)

and the expressions for the shear and bulk viscosities

µ =νT 1/2

2π1/2dG(ν)

[

1 + 2G(ν) +

(

1 +8

π

)

G(ν)2]

,(11)

λ =8νG(ν)T 1/2

π3/2d, (12)

the thermal conductivity

κ =2νT 1/2

π1/2dG(ν)

[

1 + 3G(ν) +

(

9

4+

4

π

)

G(ν)2]

, (13)

and the energy dissipation rate

ε =16νG(ν)T 3/2

π3/2d3(1 − e2). (14)

The radial pair distribution function G(ν) for a dilute2D gas of elastic hard disks can be approximated by theformula (Song et al., 1989)

GCS(ν) =ν(1 − 7ν/16)

(1 − ν)2(15)

(this is a two-dimensional analog of the famousCarnahan-Starling formula (Carnahan and Starling,1969) for elastic spheres). This formula is expected towork for densities roughly below 0.7. For high densitygranular gases, this function has been calculated usingfree volume theory by Buehler et al. (1951),

GFV =1

(1 + e)[

(νc/ν)1/2 − 1] (16)

where νc ≈ 0.82 is the density of the random close pack-ing limit. Luding (2001) proposed a global fit

GL = GCS + (1 + exp(−(ν − ν0)/m0))−1)(GFV −GCS)

with empirically fitted parameters ν0 ≈ 0.7 and m0 ≈10−2. However, even with this extension, the contin-uum theory comprised of Eqs.(5)-(14) cannot describethe force chains which transmit stress via persistent con-tacts remaining in the dense granular flows, as well asthe hysteretic transition from solid to static regimes andcoexisting solid and fluid phases.

The granular hydrodynamics is probably the mostuniversal (however not always the most appropriate)tool for modelling large-scale collective behavior indriven granular matter. Granular hydrodynamics equa-tions in the form (5),(6),(7) and their modificationsare widely used in the engineering community to de-scribe a variety of large-scale granular flows, espe-cially for design of gas-fluidized bed reactors (Gidaspow,1994). In the physics community granular hydrody-namics is used to understand various instabilities in

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relatively small-scale flows, such as flow past obstacle(Rericha et al., 2002), convection (Livne et al., 2002a,b),floating clusters (Meerson et al., 2003), longitudinal rolls(Forterre and Pouliquen, 2002, 2003), patterns in vi-brated layers (Bougie et al., 2005) and others. However,Eqs. (5)-(7) are often used far beyond their applicabil-ity limits, viz. dilute flows. Consequently, certain pa-rameters and constitutive relations need to be adjustedheuristically in order to accommodate observed behav-ior. For example, Bougie et al. (2002) had to introduceartificial non-zero viscosity in Eq. (6) for ν → 0 in or-der to avoid artificial blowup of the solution. Similarly,Losert et al. (2000) introduced the viscosity diverging asdensity approaches the close packed limit as (ν − νc)β

with β ≈ 1.75 being the fitting parameter in order todescribe the structure of dense shear granular flows.

B. Phenomenological models

A generic approach to the description of dense granu-lar flows was suggested by Aranson and Tsimring (2001,2002) who proposed to treat the shear stress mediatedfluidization of granular matter as a phase transition. Forthis purpose an order parameter characterizing the lo-cal state of granular matter and the corresponding phasefield model were introduced. According to the model, theorder parameter has its own relaxation dynamics and de-fines the static and dynamic contributions to the shearstress tensor. This approach is discussed in more detailsin Sec. VI.A.1.

Another popular approach is based on the two-phasedescription of granular flow, one phase corresponding torolling grains and the other phase to static ones. Thisapproach, so-called the BCRE model, was suggested byBouchaud et al. (1994, 1995) for description of surfacegravity driven flows. The BCRE model has direct re-lation to depth-averaged hydrodynamic equations (so-called Saint-Venant model) popular in the engineeringcommunity. Note that BCRE and Saint-Venant modelscan be derived in a certain limit from the more generalorder parameter model mentioned above, for detail seeSec. VI.A.2.

Many pattern-forming systems are often described bygeneric amplitude equations such as Ginzburg-Landau orSwift-Hohenberg equations (Aranson and Kramer, 2002;Cross and Hohenberg, 1993). This approach allows toexplain many generic features of patterns, however inany particular system there are peculiarities which needto be taken into account. This often requires modifi-cations to be introduced into the generic models. Thisapproach was taken by Aranson and Tsimring (1998);Aranson et al. (1999a); Crawford and Riecke (1999);Tsimring and Aranson (1997); Venkataramani and Ott(1998) in order to describe patterns in a vibrated gran-ular layer. Details of these approaches can be found inSec. V.D.

In addition, a variety of tools of statistical physics

are applied to diverse phenomena occurring in gran-ular systems. For example, celebrated theory ofLifshitz and Slyozov (1958) developed for coarsening

phenomena in equilibrium systems was successfully ap-plied to coarsening of clusters in granular systems(Aranson et al., 2002), see Section VIII.C.

C. Molecular dynamics simulations

Realistic simulation of granular matter consisting ofthousands of particles remains a challenge for physics andcomputer science. Due to simplicity of microscopic inter-action laws (at least for “dry” and non-cohesive granularmatter) and relatively small number of particles in gran-ular flows as compared to atomic and molecular systems,the molecular dynamics simulations or discrete elementmodels have a potential to address adequately many phe-nomena occurring in the granular systems.

There exist three fundamentally different approaches,so-called soft particles simulation method; event drivenalgorithm and the contact dynamics method for rigidparticles. For the review on various molecular dynam-ics simulation methods we recommend Luding (2004);Poschel and Schwager (2005); Rapaport (1995).

In the soft particle algorithm, all forces acting on aparticle either from walls or other particles or externalforces are calculated based on the positions of the par-ticles. Once the forces are found, the time is advancedby the explicit integration of the corresponding Newtonequations of motion. Various models are used for cal-culating normal and tangential contact forces. In ma-jority of implementations, the normal contact forces aredetermined from the particle overlap ∆n which is de-fined as the difference of the distance between the cen-ters of mass of two particles and the sum of their radii.The normal force Fn is either proportional to ∆n (lin-ear Hookian contact) or proportional to ∆3/2 (Hertziancontact). In the spring-dashpot model, additional dis-sipative force proportional to the normal component ofthe relative velocity is added to model inelasticity ofgrains. A variety of approaches are used to model tan-gential forces, the most widely accepted of them beingCundall-Strack algorithm (Cundall and Strack, 1979), inwhich the tangential contact is modelled by a dissipa-tive linear spring whose force Ft = −kt∆t − m/2γtvt

(here ∆t is the relative tangential displacement and vt

is the relative tangential velocity, kt, γt are model con-stants). It is truncated when its ratio to the normalforce |Ft|/|Fn| reaches the friction coefficient µ accord-ing to the Coloumb law. Soft-particles methods are rela-tively slow and used mostly for the analysis of dense flowswhen generally faster event-driven algorithms are notapplicable, see e.g. Landry et al. (2003); Silbert et al.

(2002a,b,c); Volfson et al. (2003a,b).In the event-driven algorithm, the particles are con-

sidered infinitely rigid and move freely (or driven bymacroscopic external fields) in the intervals between (in-

8

stantaneous) collisions. The algorithm updates veloci-ties and positions of the two particles involved in a bi-nary collision (in the simplest frictionless case, accord-ing to Eq. (1)), and then finds the time of the nextcollision and velocities and positions of all particles atthat time according to Newton’s law. Thus, the timeis advanced directly from one collision to the next, andso variable time step is dictated by the interval be-tween the collisions. While event-driven methods aretypically faster for dilute rapid granular flows, they be-come impractical for dense flows where collisions arevery frequent and furthermore particles develop persis-tent contacts. As a related numerical problem, event-driven methods are known to suffer from so-called “in-elastic collapse” when the number of collisions betweenparticles diverges in finite time (McNamara and Young,1996). There are certain modifications to this methodwhich allow to circumvent this problem by intro-ducing velocity dependent restitution coefficient (see,e.g. Bizon et al. (1998a)), but still event-driven meth-ods are mostly applied to rapid granular flows, seee.g. Ferguson et al. (2004); Khain and Meerson (2004);McNamara and Young (1996); Nie et al. (2002).

Contact dynamics is a discrete element method likesoft-particles and event-driven ones, with the equationsof motion integrated for each particle. Similarly to event-driven algorithm and unlike soft-particles method, parti-cle deformations are suppressed by considering particlesinfinitely rigid. The contact dynamics method considersall contacts occurring within a certain short time intervalas simultaneous, and computes all contact forces by sat-isfying simultaneously all kinematic constraints imposedby impenetrability of the particles and the Coulomb fric-tion law. Imposing kinematic constraints requires con-tact forces (constraint forces) which cannot be calcu-lated from the positions and velocities of particles alone.The constraint forces are determined in such a waythat constraint-violating accelerations are compensated.For comprehensive review on the contact dynamics seeBrendel et al. (2004).

Sometimes different molecular dynamics methods areoften applied to the same problem. Lois et al. (2005);Radjai et al. (1998); Staron et al. (2002) applied con-tact dynamics methods and Silbert et al. (2002a,b);Volfson et al. (2003a,b) used soft-particles techniquefor the analysis of instabilities and constitutive re-lations in dense granular systems. Patterns in vi-brated layers were studied by event-driven simulationsby Bizon et al. (1998a); Moon et al. (2003) and by softparticles molecular dynamics simulations by Nie et al.

(2000); Prevost et al. (2004).

IV. PATTERNS IN SUB-MONOLAYERS. CLUSTERING,

COARSENING AND PHASE TRANSITIONS

A. Clustering in Freely Cooling Gases

Properties of granular gases are dramatically differentfrom the properties of molecular gases due to inelastic-ity of collisions between the grains. This leads to theemergence of correlation between colliding particles andviolation of the molecular chaos approximation. Thisin turn gives rise to various pattern-forming instabili-ties. Perhaps the simplest system exhibiting nontrivialpattern formation in the context of granular matter isfreely cooling granular gas: isolated system of inelasti-cally colliding particles. The interest to freely coolinggranular gases was triggered by the discovery of clus-tering by Goldhirsch and Zanetti (1993): spontaneouslyforming dense clusters emerge as a result of instabilityof initially homogeneous cooling state, see Fig. 2. Thisinstability, which can be traced in many other granularsystems, has a very simple physical interpretation: localincrease of the density of granular gas results in the in-crease in the number of collisions, and, therefore, furtherdissipation of energy and decrease in the granular tem-perature. Due to proportionality of pressure to the tem-perature, the decrease of temperature will consequentlydecrease local pressure, which, in turn, will create a fluxof particles towards this pressure depression, and furtherincrease of the density. This clustering instability has in-teresting counterparts in astrophysics: clustering of self-gravitating gas (Shandarin and Zeldovich, 1989) and “ra-diative instability” in optically thin plasmas (Meerson,1996) resulting in interstellar dust condensation.

According to Goldhirsch and Zanetti (1993), the initialstage of clustering can be understood in terms of theinstability of a homogeneously cooled state described bythe density ν and granular temperature T . This state ischaracterized by zero hydrodynamic velocity v, and thetemperature evolution follows from the energy balanceequation

∂tT ∼ −T 3/2 (17)

which results in the Haff’s cooling law T ∼ t−2 (Haff,1983). However, the uniform cooling state becomesunstable in large enough systems masking Haff’s law.The discussion of the linear instability conditions canbe found e.g. in Babic (1993); Brilliantov and Poschel(2004). For the case of particles with fixed restitu-tion coefficient e, the analysis in the framework of lin-earized hydrodynamics equations (5)-(7) yields the criti-cal wavenumber k∗ for the clustering instability

k∗ ∼√

1 − e2 (18)

As one sees, the length scale of the clustering instabilitydiverges in the limit of elastic particles e→ 1.

The clustering instability in a system of grains withconstant restitution coefficient results in the inelastic

9

collapse discussed in the previous Section. Whereasthe onset of clustering can be well-understood inthe framework of granular hydrodynamics (see, e.g.Babic (1993); Brilliantov and Poschel (2004); Goldhirsch(2003); Hill and Mazenko (2003)), certain subtle features(e.g. scaling exponents for temperature) are only as-sessed within molecular dynamics simulation because thehydrodynamic description often breaks in dense cold clus-ters. One recent theoretical approach to the descriptionof the late stages of clustering instability consists in intro-ducing additional regularization into the hydrodynamicdescription due to the finite size of particles (Efrati et al.,2005; Nie et al., 2002).

Nie et al. (2002) argued that cluster formation and co-alescence in freely cooling granular gases can be heuris-tically described by the Burgers equation for hydrody-namic velocity v with random initial conditions:

∂tv + v∇v = µ0∇2v (19)

where µ0 is effective viscosity (which is different from theshear viscosity in hydrodynamic equations). In this con-text clustering is associated with the formation of shocksin the Burgers equation. Perhaps not surprising, a verysimilar approach was applied for description of the gasof “sticky” particles for the description of the large-scalematter formation in the Universe (Gurbatov et al., 1985;Shandarin and Zeldovich, 1989).

Meerson and Puglisi (2005) conducted molecular dy-namics simulations of the clustering instability of a freelycooling dilute inelastic gas in a quasi-one-dimensional set-ting. This problem was also examined in the frameworkof granular hydrodynamics by Efrati et al. (2005). It wasobserved that, as the gas cools, stresses become negligiblysmall, and the gas flows only by inertia. Hydrodynamicdescription reveals a finite-time singularity, as the veloc-ity gradient and the gas density diverge at some location.The molecular dynamics studies show that finite-timesingularities, intrinsic in such flows, are arrested onlywhen close-packed clusters are formed. It was confirmedthat the late-time dynamics and coarsening behavior aredescribable by the Burgers equation (19) with vanishingviscosity µ0. Correspondingly, the average cluster massgrows as t2/3 and the average velocity decreases as t−1/3.Due to the clustering long-term temperature evolution isT ∼ t−2/3 which is different from Haff’s law T ∼ t−2 de-rived for the spatially-homogeneous cooling. Efrati et al.

(2005) argue that flow by inertia represents a generic in-termediate asymptotic of unstable free cooling of dilutegranular gases consistent with the Burgers equation (19)description of one-dimensional gas of “sticky particles”suggested by Nie et al. (2002).

While there is a qualitative similarity between Burgersshocks and clusters in granular materials at least in onedimension, the applicability of the Burgers equation forthe description of granular media is still an open ques-tion, especially in two and three dimensions. The mainproblem is that the Burgers equation can be derived fromthe hydrodynamic equations only in one dimensional sit-

uation, in two and three dimensions the Burgers equationassumes zero vorticity, which possibly oversimplifies theproblem and may miss important physics. In fact, molec-ular dynamics simulations illustrate the development oflarge-scale vortex flows in the course of clustering insta-bility (Catuto and Marconi, 2004; van Noije and Ernst,2000).

Finally, Das and Puri (2003) proposed a phenomeno-logical description of the long-term clusters evolution ingranular gases. Using the analogy between clusteringin granular gases and phase-ordering dynamics in two-component mixtures, Das and Puri (2003) postulatedgeneralized Cahn-Hilliard equations for the evolution ofdensity ν and complex velocity ψ = vx + ivy (see e.g.(Bray, 1994))

∂tν = (−∇2)m[

ν − ν3 + ∇2ν]

(20)

∂tψ = (−∇2)m[

ψ − |ψ|2ψ + ∇2ψ]

(21)

with m → 0+ which characterize globally-conserved dy-namics of ν and ψ similar to that considered in Sec.VIII.C.1. Das and Puri (2003) argue that this choiceis most appropriate due to the non-diffusive characterof particles motion and is consistent with the observedmorphology of clusters. While it might be very challeng-ing to derive Eqs. (20),(21) from the first principles orto deduce them from hydrodynamic equations, the con-nection to phase-ordering dynamics is certainly deservesfurther investigation.

B. Patterns in Driven Granular Gases

Discovery of the clustering instability stimulated alarge number of experimental and theoretical studies,even experiments in low gravity conditions (Falcon et al.,1999). Since “freely cooling granular gas” is difficult toimplement in the laboratory, most experiments were per-formed in the situation when the energy is injected in thegranular system in one or another way. Kudrolli et al.

(1997) studied two-dimensional granular assemblies in-teracting with a horizontally vibrating (or “hot”) wall.In agreement with granular hydrodynamics, maximumgas density occurs opposite to the vibrating wall, seeFig. 16. The experimental density distributions are con-sistent with the modified hydrodynamic approach pro-posed by Grossman et al. (1997). Khain and Meerson(2002); Khain et al. (2004a); Livne et al. (2002a,b), stud-ied the dynamics of granular gases interacting with a hotwall analytically using granular hydrodynamic theory forrigid disks in the formulation of Jenkins and Richman(1985) and predicted a novel phase-separation or van derWaals-type instability of the one-dimensional density dis-tribution. This instability, reproduced later by molecu-lar dynamics simulations (Argentina et al., 2002) is dif-ferent from the usual convection instability as it occurswithout gravity and is driven by the coarsening mecha-nism. Simulations indicated a profound role of fluctua-tions. One may expect that noise amplification near the

10

FIG. 16 Sample image showing dense cold cluster formed op-posite the driving wall (at the bottom), total number of par-ticles 1860, from Kudrolli et al. (1997).

instability thresholds in granular systems will be veryimportant due to non-macroscopic number of grains. Inthe context of phase-separation instability Meerson et al.

(2004) raised the non-trivial question of the origin of gi-ant fluctuations and break-down of hydrodynamic de-scription in granular systems near the threshold of in-stability (see also (Bougie et al., 2005; Goldman et al.,2004) on the effect of fluctuations in multilayers). Re-markably, for the granular gas confined between two os-cillating walls Khain and Meerson (2004) predicted onthe basis of event-driven simulations a novel oscillatoryinstability for the position of the dense cluster. Thesepredictions, however, have not yet been confirmed exper-imentally, most likely due to the relatively small aspectratio of available experimental cells.

Olafsen and Urbach (1998) pioneered experimentswith sub-monolayers of particles subject to vertical vibra-tion2. Their studies revealed a surprising phenomenon:formation of a dense closely-packed cluster co-existingwith dilute granular gas, see Fig. 1. The phenomenonbears a strong resemblance to the first-order solid/liquidphase transition in equilibrium systems. Similar ex-periments by Losert et al. (1999) discovered propagatingfronts between gas-like and solid-like phases in verticallyvibrated sub-monolayers. Such fronts are expected inextended systems in the vicinity of the first order phasetransition, e.g. solidification fronts in supercooled liq-uids. Prevost et al. (2004) performed experiments withvibrated granular gas confined between two plates. Qual-itatively similar phase coexistence was found. The clusterformation in vibro-fluidized sub-monolayers shares manycommon features with processes in freely cooling gran-ular gases because it is also caused by the energy dissi-pation due to inelasticity of collisions. However, there isa significant difference: the instability described in Sub-sec. IV.A is insufficient to explain the phase separation.A very important additional factor is bistability and co-existence of states due to the nontrivial density depen-dence of the transfer rate of particle’s vertical to horizon-tal momentum. Particles in a dense closed-packed clusterlikely obtain less horizontal momentum than in a moder-ately dilute gas because in the former particle vibrationsare constrained to the vertical plane by interaction withneighbors. In turn, in a very dilute gas the vertical tohorizontal momentum transfer is also inhibited due tolack of particle collisions. Another factor here is that vi-bration is not fully equivalent to the interaction with aheat bath. It is well known that even a single particle

2 Sub-monolayer implies less than 100% percent coverage by par-ticles of the bottom plate.

interacting with a periodically vibrating plate exhibitscoexistence of dynamic and static states (Losert et al.,1999).

There were several simulation studies of clusteringand phase coexistence in vibrated granular submono-layers. Nie et al. (2000); Prevost et al. (2004) repro-duced certain features of cluster formation and two-phaseco-existence by means of large-scale three-dimensionalmolecular dynamics simulations. Since realistic three-dimensional simulations are still expensive and extremelytime-consuming, simplified modelling of the effect of avibrating wall by a certain multiplicative random forc-ing on individual particles was employed by Cafiero et al.

(1999). While the multiplicative random forcing is an in-teresting theoretical idea, it has to be used with cautionas it is not guaranteed to reproduce subtle details of par-ticle dynamics, especially the sensitive dependence of thevertical to horizontal momentum transfer as the functionof the density.

C. Coarsening of clusters

One of the most intriguing questions in the con-text of phase coexistence in vibrofluidized granularsub-monolayers is a possibility of Ostwald-type ripen-ing and coarsening of clusters similar to that ob-served in equilibrium systems ( Lifshitz and Slyozov,1958, 1961). In particular, the scaling law for the num-ber of macroscopic clusters is of special interest becauseit gives a deep insight into the similarity between equi-librium thermodynamic systems and non-equilibriumgranular systems. The experiments (Losert et al.,1999; Olafsen and Urbach, 1998; Prevost et al., 2004;Sapozhnikov et al., 2003) demonstrated emergence andgrowth of multiple clusters but did not have sufficientaspect ratio to address the problem of coarsening in aquantitative way.

Nevertheless, as it was suggested by Aranson et al.

(2000), statistical information on out-of-equilibrium Ost-wald ripening can be obtained in a different granular sys-tem: electrostatically driven granular media. This sys-tem permits one to operate with extremely small par-ticles and obtain a very large number of macroscopicclusters. In this system the number of clusters N de-cays with time as N ∼ 1/t. This law is consistentwith interface-controlled Ostwald ripening in two di-mensions, see (Wagner, 1961). Whereas mechanismsof energy injections are different, both vibrofluidizedand electrostatically-driven systems show similar behav-ior: macroscopic phase separation, coarsening, transi-tion from two- to three-dimensional cluster growth, etc(Sapozhnikov et al., 2003). In Aranson et al. (2002) thetheoretical description of granular coarsening was devel-oped in application to the electrostatically driven grains,however we postpone the description of this theory toSec. VIII.C. We anticipate that a theory similar to thatformulated in Aranson et al. (2002) can be applicable to

11

mechanically fluidized granular materials as well. Themain difference there is the physical mechanism of en-ergy injection which will possibly affect the specific formof the conversion rate function φ in Eq. (69) in Sec.VIII.C.

V. SURFACE WAVES AND PATTERNS IN VIBRATED

MULTILAYERS OF GRANULAR MATERIALS

A. Chladni patterns and heaping

Driven granular systems often manifest collective fluid-like behavior: shear flows, convection, surface waves, andpattern formation (see e.g. Jaeger et al. (1996)). Sur-prisingly, even very thin (less than ten) layers of sandunder excitation exhibit pattern formation which is quitesimilar (however with some important differences) to thecorresponding patterns in fluids. One of the most fasci-nating examples of these collective dynamics is the ap-pearance of long-range coherent patterns and localizedexcitations in vertically-vibrated thin granular layers.

Experimental studies of vibrated layers of sand have along and illustrious history, beginning from the seminalworks by Chladni (1787) and Faraday (1831) in whichthey used a violin bow and a membrane to excite verticalvibrations in a thin layer of grains. The main effect ob-served in those early papers, was “heaping” of granularmatter in mounds near the nodal lines of the membraneoscillations. This behavior was immediately (and cor-rectly) attributed to the “acoustic streaming”, or nonlin-ear detection of the nonuniform excitation of grains bymembrane modes. One puzzling result by Chladni wasthat a very thin powder would collect at the anti-nodalregions where the amplitude of vibrations is maximal. AsFaraday demonstrated by evacuating the container, thisphenomenology is caused by the role of air permeatingthe grains in motion. Evidently, the interstitial gas be-comes important as the terminal velocity of a free fallvt = νgd2/18µ becomes of the order of the plate velocity,and this condition is fulfilled for 10− 20 µm particles ona plate vibrating with frequency 50 Hz and accelerationamplitude g.

In subsequent years the focus of attention was di-verted from dynamical properties of thin layers of vi-brated sand, and only in the last third of the 20th cen-tury physicists returned to this old problem equippedwith new experimental capabilities. The dawn of thenew era was marked by the studies of heaping by Jenny(1964). In subsequent papers (Dinkelacker et al., 1987;Douady et al., 1989; Evesque et al., 1989; Laroche et al.,1989; Walker, 1982), more research has been performed ofheaping with and without interstitial gas, with somewhatcontroversial conclusions on the necessity of ambient gasfor heaping (see, e.g. (Evesque, 1990)). Eventually, aftermore careful analysis Pak et al. (1995) concluded thatheaping indeed disappears as the pressure of the am-bient gas tends to zero or the particle size increases.

This agreed with numerical molecular dynamics simu-lations (Gallas et al., 1992a,b,c; Gallas and Soko lowski,1993; Luding et al., 1994; Taguchi, 1992) which showedno heaping without interstitial gas effects. Recent stud-ies of deep layers (50 < N < 200) of small particles(10 < d < 200µm) by Duran (2000, 2001); Falcon et al.

(1999a) showed a number of interesting patterns andnovel instabilities caused by interstitial air. In particu-lar, Duran (2001) observed formation of isolated dropletsof grains after periodic taping similar to the Rayleigh-Taylor instability in ordinary fluids.

Jia et al. (1999) proposed a simple model for heap for-mation which is motivated by these experiments. In adiscrete lattice version of the model, the decrease in lo-cal density due to vibrations is modelled by the randomcreation of empty sites in the bulk. The bulk flow is sim-ulated by the dynamics of empty sites, while the surfaceflow is modelled by rules similar to the sandpile model(see Sec. VI.B). This model reproduced both convectioninside the powder and the heap formation for sufficientlylarge probability of empty site formation (which mimicsthe magnitude of vibration). Jia et al. (1999) also pro-posed the continuum model which has a simple form of anonlinear reaction-diffusion equation, for the local heightof the sandpile

∂th = D∇2h+ Ωh− βh2. (22)

However this model is perhaps too generic and lacks thespecific physics of the heaping process.

B. Standing wave patterns

While heaping may or may not appear depending onthe gas pressure and the particle properties at small ver-tical acceleration, at higher vertical acceleration patternsof standing waves emerge in thin layers. They were firstreported by Douady et al. (1989); Fauve et al. (1989) ina quasi two-dimensional geometry. These waves oscil-lated at the half of the driving frequency, which indi-cates the sub-harmonic resonance characteristic for para-metric instability. This first observation spurred a num-ber of experimental studies of standing waves in thingranular layers in two and three dimensional geometries(Aranson et al., 1999b; Clement et al., 1996; Melo et al.,1994, 1995; Mujica and Melo, 1998; Umbanhowar et al.,1996). Importantly, these studies were performed in evac-uated containers, which allowed to obtain reproducibleresults not contaminated by heaping. Fig. 3 shows avariety of regular patterns observed in vibrated granularlayers under vibration (Melo et al., 1994). As a result ofthese studies, the emerging picture of pattern formationappears as follows.

The particular pattern is determined by the interplaybetween driving frequency f and acceleration of the con-tainer Γ = 4π2Af2/g (A is the amplitude of oscillations,g is the gravity acceleration) (Melo et al., 1994, 1995).The layer of grains remains flat for Γ < 2.4 more-less

12

FIG. 17 Phase diagram of various regimes in vibrated gran-ular layers, from Melo et al. (1995).

independent of driving frequency. At higher Γ patternsof standing waves emerge. At small frequencies f < f∗

(for experimental conditions of Melo et al. (1995) , f∗ ≈45 Hz) the transition is subcritical, leading to the for-mation of square wave patterns, see Fig.3b. For higherfrequencies f > f∗ the selected pattern is quasi-one-dimensional stripes (Fig. 3a), and the transition be-comes supercritical. In the intermediate region f ∼ f∗,localized excitations (oscillons, Fig.4) and various boundstates of oscillons (Fig.3f) were observed within the hys-teretic region of the parameter plane. Both squares andstripes, as well as oscillons, oscillate at the half of thedriving frequency, which indicates the parametric mech-anism of their excitation. The wavelength of the cellularpatterns near the onset scales linearly with the depth ofthe layer and diminishes with the frequency of vibration(Umbanhowar and Swinney, 2000). The frequency cor-responding to the strip-square transition was shown todepend on the particle diameter d as d−1/2. This scalingsuggests that the transition is controlled by the relativemagnitude of the energy influx from the vibrating plate∝ f2 and the gravitational dilation energy ∝ gd. Athigher acceleration (Γ > 4), stripes and squares becomeunstable, and hexagons appear instead (Fig. 3c). Furtherincrease of acceleration at Γ ≈ 4.5 converts hexagons intoa domain-like structure of flat layers oscillating with fre-quency f/2 with opposite phases. Depending on parame-ters, interfaces separating flat domains, are either smoothor “decorated” by periodic undulations (Fig. 3e). ForΓ > 5.7 various quarter-harmonic patterns emerge. Thecomplete phase diagram of different regimes observed ina three-dimensional container is shown in Fig. 17. Foreven higher acceleration (Γ > 7) the experiments revealsurprising phase bubbles and spatio-temporal chaos oscil-lating approximately at one fourth the driving frequency(Moon et al., 2002).

Subsequent investigations revealed that periodic pat-terns share many features with convective rolls inRayleigh-Benard convection, for example skew-varicoseand cross-roll instabilities (de Bruyn et al., 1998).

C. Simulations of vibrated granular layers

The general understanding of the standing wave pat-terns in thin granular layers can be gained by the anal-ogy with ordinary fluids. The Faraday instability in flu-ids and corresponding pattern selection problems havebeen studied theoretically and numerically in great de-tail (see e.g. Zhang and Vinals (1997)). The primarymechanism of instability is the parametric resonance be-tween the spatially uniform periodic driving at frequencyf and two counter-propagating gravity waves at fre-quency f/2. However, this instability in ordinary flu-

FIG. 18 Comparison between subharmonic patterns in exper-iment (left) and three dimensional molecular dynamics sim-ulations (right) of 30000 particles in a square vibrated con-tainer for different frequencies and amplitudes of vibration,from Bizon et al. (1998a).

ids leads to a supercritical bifurcation and square wavepatterns near offset, and as a whole the correspondingphase diagram lacks the richness of the granular sys-tem. Of course this can be explained by the fact thatthere are many qualitative differences between granu-lar matter and fluids, such as presence of strong dis-sipation, friction and the absence of surface tension inthe former. Interestingly, localized oscillon-type objectswere subsequently observed in vertically vibrated lay-ers of non-Newtonian fluid (Lioubashevski et al., 1999),and stipe patterns were observed in highly viscous fluid(Kiyashko et al., 1996). The theoretical understandingof the pattern formation in a vibrated granular systempresents a challenge, since unlike fluid dynamics thereis no universal theoretical description of dense granularflows analogous to the Navier-Stokes equations. In theabsence of this common base, theoretical and computa-tional efforts in describing these patterns followed severaldifferent directions. Aoki et al. (1996) were first to per-form molecular dynamics simulations of patterns in thevibrated granular layer. They concluded that grain-grainfriction is necessary for pattern formation in this system.However, as noted by Bizon et al. (1997), this conclusionis a direct consequence of the fact that the algorithm ofAoki et al. (1996), which is based on the Lennard-Jonesinteraction potential and velocity dependent dissipation,leads to the restitution coefficient of particles approach-ing unity for large collision speeds rather than decreasingaccording to experiments.

Bizon et al. (1998a,b) performed event-driven simula-tions of colliding grains on a vibrated plate assuming con-stant restitution (see also Luding et al. (1996) for ear-lier two-dimensional event-driven simulations). It wasdemonstrated that even without friction, patterns doform in the system, however only supercritical bifurca-tion to stripes is observed. It turned out that frictionis necessary to produce other patterns observed in ex-periments, such as squares and f/4 hexagons. Simula-tions with frictional particles reproduced the majority ofpatterns observed in experiments and many features ofthe bifurcation diagram (with the important exception ofthe oscillons). Bizon et al. (1998a) set out to match anexperimental cell and a numerical system, maintainingexactly the same size container and sizes and the num-ber of particles. After fitting only two parameters of thenumerical model, Bizon et al. (1998a) were able to finda very close quantitative agreement between various pat-terns in the experimental cell and patterns in simulationsthroughout the parameter space of the experiment (fre-quency of driving, amplitude of acceleration, thickness ofthe layer), see Fig. 18.

13

Shinbrot (1997) proposed a model which combinedideas from molecular dynamics and continuum mod-elling. Specifically, the model ignored vertical compo-nent of particle motion and assumed that impact withthe plate adds certain randomizing horizontal velocity tothe individual particles. The magnitude of the randomcomponent being added at each impact served as a mea-sure of impact strength. After the impact particles wereallowed to travel freely in the horizontal plane for a cer-tain fraction of a period after which they inelasticallycollide with each other (a particle acquires momentumaveraged over all particles in its neighborhood). Thismodel did reproduce a variety of patterns seen in experi-ments (stripes, squares, and hexagons) for various valuesof control parameters (frequency of driving and impactstrength), however it did not describe some of the ex-perimental phenomenology (localized objects as well asinterfaces), besides it also produced a number of intricatepatterns not seen in experiments.

D. Continuum theories

The first continuum models of pattern formationin vibrating sand were purely phenomenological. Inthe spirit of weakly-nonlinear perturbation theoriesTsimring and Aranson (1997) introduced the complexamplitude ψ(x, y, t) of sub-harmonic oscillations of thelayer surface, h = ψ exp(iπft) + c.c.. The equation forthis function on the symmetry grounds in the lowest or-der was written as

∂tψ = γψ∗− (1− iω)ψ+(1+ ib)∇2ψ−|ψ|2ψ−νψ. (23)

Here γ is the normalized amplitude of forcing at the driv-ing frequency f . The linear terms in Eq. (23) can beobtained from the complex growth rate for infinitesimalperiodic layer perturbations h ∼ exp[Λ(k)t + ikx]. Ex-panding Λ(k) for small k, and keeping only two leadingterms in the expansion Λ(k) = −Λ0 − Λ1k

2 gives riseto the linear terms in Eq. (23), where b = ImΛ1/ReΛ1

characterizes ratio of dispersion to diffusion and parame-ter ω = −(ImΛ0+πf)/ReΛ0, characterizes the frequencyof the driving.

The only difference between this equation and theGinzburg-Landau equation for the parametric instabil-ity (Coullet et al., 1990) is the coupling of the complexamplitude ψ to the “slow mode” ν which characterizeslocal dissipation in the granular layer (ν can be inter-preted as coarse-grained layer’s number density). Thisslow mode obeys its own dynamical equation

∂tν = α∇ · (ν∇|ψ|2) + β∇2ν. (24)

This equation describes re-distribution of the averagedthickness due to the diffusive flux ∝ −∇ν, and an addi-tional flux ∝ −ν∇|ψ|2 is caused by the spatially nonuni-form vibrations of the granular material. This cou-pled model was used by Aranson and Tsimring (1998);

FIG. 19 Phase diagram showing primary stable patterns de-rived from Eqs. (23),(24). Points indicate stable oscillonsobtained by numerical solution of Eqs. (23),(24), η = α/β,µ = 〈ν〉 is average density, and ǫ ∼ γ − γc is supercriticalityparameter, from Tsimring and Aranson (1997).

FIG. 20 Radially-symmetric oscillon solution ofEqs.(23),(24) for γ = 1.8, µ = 0.567, b = 2, ω = α =1, η = 5/γ, from Tsimring and Aranson (1997).

Tsimring and Aranson (1997) to describe the patternselection near the threshold of the primary bifurcation.The phase diagram of various patterns found in thismodel is shown in Fig. 19. At small α〈ν〉β−1 (which cor-responds to low frequencies and thick layers), the primarybifurcation is subcritical and leads to the emergence ofsquare patterns. For higher frequencies and/or thinnerlayers, transition is supercritical and leads to roll pat-terns. At intermediate frequencies stable localized solu-tions of Eqs.(23),(24) corresponding to isolated oscillons

and a variety of bound states were found in agreementwith experiment. The mechanism of oscillon stabiliza-tion is related to the oscillatory asymptotic behavior ofthe tails of the oscillon (see Fig. 20), since this under-lying periodic structure provides pinning for the circularfront forming the oscillon. Without such pinning, the os-cillon solution could only exist at a certain unique valueof a control parameter (e.g. γ), and would either collapseor expand otherwise.

Let us note that stable localized solutions somewhat re-sembling oscillons have recently been found in the nonlin-ear Schrodinger equation with additional linear dissipa-tion and parametric driving (Barashenkov et al., 2002).

Phenomenological model (23),(24) also provides a gooddescription of patterns away from the primary bifurca-tion - hexagons and interfaces (Aranson et al., 1999a).In high-frequency limit the slow mode dynamics can beneglected (ν becomes enslaved by ψ), and the dynam-ics can me described by a single parametric Ginzburg-Landau equation (23).

It is convenient to shift the phase of the complex orderparameter via ψ = ψ exp(iφ) with sin 2φ = ω/γ. The

equations for real and imaginary part ψ = A+ iB are:

∂tA = (s− 1)A− 2ωB − (A2 +B2)A+ ∇2(A− bB),

∂tB = −(s+ 1)B − (A2 +B2)B + ∇2(B + bA), (25)

where s2 = γ2−ω2. At s < 1, Eqs. (25) has only one triv-ial uniform state A = 0, B = 0, At s > 1, two new uni-form states appear, A = ±A0, B = 0, A0 =

√s− 1. The

onset of these states corresponds to the period doublingof the layer flights sequence, observed in experiments(Melo et al., 1994, 1995) and predicted by the simpleinelastic ball model (Mehta and Luck, 1990; Melo et al.,1994, 1995). Signs ± reflect two relative phases of layerflights with respect to container vibrations.

14

Weakly-nonlinear analysis reveals that the uniformstates ±A0 lose their stability with respect to finite-wavenumber perturbations at s < sc, and the nonlin-ear interaction of growing modes leads to hexagonal pat-terns. The reason for this is that the non-zero base stateA = ±A0 lacks the up-down symmetry ψ → −ψ and thecorresponding amplitude equations contains quadraticterms which are known to favor hexagons close to onset(see, e.g. Cross and Hohenberg (1993)). In the regimewhen the uniform states A = ±A0, B = 0 are stable,there is an interface solution connecting these two asymp-totic states. This interface may exhibit transversal in-stability which leads to decorated interfaces (see exper-imental Fig. 3e). Due to symmetry, the interfaces areimmobile, however breaking the symmetry of driving canlead to interface motion. This symmetry breaking can beachieved by additional subharmonic driving at frequencyf/2. The interface will move depending on the relativephases of f and f/2 harmonics of driving. This interfacedrift was predicted in (Aranson et al., 1999a) and ob-served in the subsequent work (Aranson et al., 1999b).As it was noted by Aranson et al. (1999b) (see also laterwork by Moon et al. (2003)), moving interfaces can beused to separate granular material of different sizes. Thestability and transition between flat and decorated in-terfaces was studied theoretically and experimentally byBlair et al. (2000). It was shown that non-local effectsare responsible for the saturation of transverse insta-bility of interfaces. Moreover, new localized solutions(“superoscillons”) were found for large accelerations. Incontrast with conventional oscillons existing on the flatbackground oscillating with driving frequency f , i.e. inour notation ψ = 0, the superoscillons exist on the back-ground of the flat period-doubled solution ψ 6= 0.

Another description of the primary pattern-forming bi-furcation was done by Crawford and Riecke (1999) inthe framework of the generalized Swift-Hohenberg equa-tion

∂tψ = Rψ − (∂2x + 1)2ψ + bψ3 − cψ5 + ε∇ · [(∇ψ)3]

− β1ψ(∇ψ)2 − β2ψ2∇2ψ. (26)

Here the (real) function ψ characterizes the amplitude ofthe oscillating solution, so implicitly it is assumed thatthe whole pattern always oscillates in phase. Terms pro-portional to ε have been added to the standard Swift-Hohenberg equation first introduced for description ofconvective rolls (see, e.g. Cross and Hohenberg (1993))since they are known to favor square patterns, and ex-tended fifth-order local nonlinearity allowed to simulatesubcritical bifurcation for R < 0. This equation also de-scribes both square and stripe patterns depending on themagnitude of ε and for negative R has a stable oscillon-type solution.

Even more generic approach was taken byVenkataramani and Ott (1998, 2001) who arguedthat the spatio-temporal dynamics of patterns gener-ated by parametric forcing can be understood in theframework of a discrete-time, continuous space system

which locally exhibits a sequence of period-doublingbifurcations and whose spatial coupling operator selectsa certain spatial scale. In particular they studied thediscrete-time system

ξn+1(x) = L[M(ξn(x))] (27)

where local mapping M(ξ) is described by a Gaussianmap

M(ξ) = r exp[−(ξ − 1)2/2]

and the linear spatial operator L has an azimuthally sym-metric Fourier transform

f(k) = sign[k2c − k2] exp[k2(1 − k2/2k2

0))/2].

Here k is the wavenumber, kc, k0 are two inverse lengthscales characterizing the spatial coupling, and r describesthe amplitude of forcing. While this choice of the spa-tial operator appears rather arbitrary, it leads to a phasediagram on the plane (kc/k0, r) which is similar to theexperimental one.

Several authors (Cerda et al., 1997;Eggers and Riecke, 1999; Park and Moon, 2002) at-tempted to develop a quasi two-dimensional fluid-dynamics-like continuum description of the vibratedsand patterns. These models deal with mass andmomentum conservation equations which are augmentedby specific constitutive relations for the mass flux andpressure. Cerda et al. (1997) assumed that during im-pact particles acquire horizontal velocities proportionalto the gradient of local thickness, then during the flightthat move freely with these velocities and redistributemass, and during the remainder of the cycle the layerdiffusively relaxes on the plate. The authors found thata flat layer is unstable with respect to square patternformation, however the transition is supercritical. Inorder to account for the subcritical character of theprimary bifurcation to square patterns, the authors pos-tulated the existence of a certain critical slope (relatedto the repose angle) below which the free flight initiatedby the impact does not occur. They also observed theexistence of localized excitations (oscillons and boundstates), however they appeared only as transients inthe model. Park and Moon (2002) generalized thismodel by explicitly writing the momentum conservationequation and introducing the equation of state for thehydrodynamic pressure which is proportional to thesquare of the velocity divergence. This effect providessaturation of the free-flight focusing instability and leadsto a squares-to-stripes transition at higher frequencieswhich was missing in the original model (Cerda et al.,1997). By introducing multiple free-flight times andcontact times Park and Moon (2002) were also able toreproduce hexagonal patterns and superlattices.

Full three-dimensional continuum simulations based onthe granular hydrodynamics equations (5),(6),(7) wereperformed by Bougie et al. (2005). Quantitative agree-ment was found between this description and event-driven molecular dynamics simulations and experiments

15

FIG. 21 Dispersion relation for stipes near the onset ac-cording to continuum granular hydrodynamics equations andmolecular dynamics simulations compared with experimentaldata, from Bougie et al. (2005).

in terms of the wavelength dependence on the vibrationfrequency (Fig.21) although the authors had to introducea certain regularization procedure in the hydrodynamicequations in order to avoid artificial numerical instabil-ities for ν → 0. Since standard granular hydrodynam-ics does not take into account friction among particles,the simulations only yielded stripe pattern, in agreementwith earlier molecular dynamics simulations. Further-more, the authors found a small but systematic difference(∼10%) between the critical value of plate accelerationin fluid-dynamical and molecular dynamics simulationswhich could be attributed to the role of fluctuations nearthe onset. Proper account of inter-particle friction andfluctuations within the full hydrodynamics descriptionstill remains an open problem (see more on that in Sec-tion VI).

Fluctuations are expected to play a significantlygreater role in granular hydrodynamics than in usualfluids, because the total number of particles involvedin the dynamics per characteristic spatial scale of theproblem is many orders of magnitude smaller than theAvogadro number. The apparatus of fluctuating hy-drodynamics which was developed in particular for de-scription of transition to rolls in Rayleigh-Benard con-vection (Swift and Hohenberg, 1977), has been recentlyapplied to the granular patterns (Bougie et al., 2005;Goldman et al., 2004). The Swift-Hohenberg theory isbased on the equation for the order parameter ψ,

∂tψ = [ǫ− (∇2 + k20)2]ψ − ψ3 + η(x, t), (28)

where ǫ is the bifurcation parameter, k0 is the wavenum-ber corresponding to the most unstable perturbations,and η is the Gaussian δ correlated noise term with in-tensity F . The Swift-Hohenberg theory predicts thatnoise offsets the bifurcation value of the control param-eter from the mean-field value ǫMF = 0 to the criticalvalue ǫc ∝ F 2/3. Furthermore, the Swift-Hohenberg the-ory describes the transition to the linear regime whichis expected to work far away from the bifurcation pointfor small noise intensity when the magnitude of noise-excited modes scales as |ǫ − ǫc|−1/2, while the time co-herence of fluctuations and the amplitude of spectralpeaks decays as |ǫ − ǫc|−1. Fitting the Swift-Hohenbergequation (28) to match the transition in vibrated gran-ular layer, Bougie et al. (2005); Goldman et al. (2004)found a good agreement with molecular dynamics sim-ulations and experiments (see, e.g., Fig.22). Interest-ingly, the magnitude of the fitted noise term in Eq.(28)F ≈ 3.5 · 10−3 turned out to be an order of magnitudegreater than for convective instability in a fluid near acritical point (Oh and Ahlers, 2003). This discrepancycould stem from the fact that the Swift-Hohenberg the-

FIG. 22 Comparison between the Swift-Hohenberg theoryand experiment for noise peak intensity (a), total noise power(b) and the correlation time (c). Symbols - experiment, solidlines - Swift-Hohenberg theory, dashed lines - linear theoryfor small noise magnitude, from Goldman et al. (2004).

FIG. 23 Select patterns in a shallow fluidized bed with pe-riodically modulated air flow for different flow parameters,from Li et al. (2003)

ory, developed for ordinary fluids, is formally valid forthe second-order phase transition, whereas in granularsystem the transition to square patterns is of the firstorder type. Consequently, the nonlinear terms can beimportant near the transition point and may distort thescaling for the noise amplitude.

There have been attempts to connect patterns in vi-brated layers with the phenomenon of granular “ther-moconvection”. Since high-frequency vibration in manyaspects is similar to “hot” wall, it was argued that oneshould expect granular temperature gradients, densityinversion, and, consequently convection instability simi-lar to that observed in heated from below liquid layers.The theoretical analysis based on granular hydrodynamicequations (5),(6),(7) supports the existence of a convec-tive instability in a certain range of parameters (He et al.,2002; Khain and Meerson, 2003). Multiple convec-tion roles were observed in molecular dynamics sim-ulations (Paolotti et al., 2004; Sunthar and Kumaran,2001). However, the experiments are not conclusiveenough (Wildman et al., 2001). In particular it appearsvery hard to discriminate between convection induced byvibration and convective flows induced by walls, see e.g.(Garcimarin et al., 2002; Pak and Behringer, 1993).

Vibrated bottom plate is not the only way to induceparametric patterns in thin granular layers. Li et al.

(2003) demonstrated that periodically modulated airflowthrough a shallow fluidized bed also produces interestingpatterns in the granular layer which oscillate at half thedriving frequency (Fig. 23). While the physical mech-anism of interaction between the airflow and grains isquite different from the collisional energy transfer in vi-brated containers, phenomenological models based on theprincipal symmetry of the problem should be able to de-scribe the gas-driven granular layer as well. In case ofthe parametric Ginzburg-Landau model Eq. (23), theorder parameter would correspond to the amplitude ofthe subharmonic component of the surface deformation,and the driving term would be related to the amplitudeof the flow modulation. Moreover, variations of the meanflow rate act similar to the variations of the gravitationalacceleration in the mechanical system, which may givean additional means to control the state of the system.

16

FIG. 24 Stability diagram for avalanches in thin granularlayers, se also Fig. 6, from Daerr and Douady (1999)

VI. PATTERNS IN GRAVITY-DRIVEN DENSE

GRANULAR FLOWS

In this Section we overview theoretical models for var-ious pattern-forming instabilities in dense gravity-drivengranular flows.

A. Avalanches in thin granular layers

Gravity-driven particulate flows are a common oc-currence in nature (dune migration, erosion/depositionprocesses, land slides, underwater gravity currents andcoastal geomorphology) and in various industrial ap-plications having to do with handling granular materi-als, including their storage, transport, and processing.One of the most spectacular (and often very dangerous)forms of gravity-driven granular flows is the avalanche.Avalanches occur spontaneously when the slope of thegranular material exceeds a certain angle (static angle ofrepose) or they can be initiated at somewhat smaller an-gle by applying a finite perturbation. Laboratory studiesof avalanches are often carried out in rotating drums (seebelow) or in a chute geometry when a layer of sand is ti-tled at a certain fixed angle. Daerr and Douady (1999)conducted experiments with a thin layer of granular mat-ter on sticky (velvet) inclined plane, see Fig. 24. Sur-prising diversity of avalanche behavior was observed inthis seemingly simple system: triangular avalanches de-veloped in thin layers (h is the layer thickness) and forsmall inclination angles φ, whereas in thicker layers orsteeper angles φ the avalanches assumed balloon shapedwith upper edge of the avalanche propagating up-hill, seeFig. 6. According to Rajchenbach (2002a, 2003) the rearfront of the balloon-like avalanche propagates uphill withthe velocity roughly one half of the downhill velocity ofthe head front, and the velocity of the head is also twotimes larger than the depth-averaged flow velocity. Thestability diagram is outlined in Fig. 24: a granular layeris stable below solid line (so-called hstop limit accordingto Pouliquen (1999)), spontaneous avalanching was ob-served above the dashed line. Between dashed and solidlines the layer exhibits bistable behavior: finite pertur-bation can trigger an avalanche, otherwise the layer re-mains stable. The dotted line with ×-symbols indicatesthe transition between triangular and balloon avalanches.

1. Partially fluidized flows

The avalanche dynamics described above is an exam-ple of a wide class of partially fluidized granular flows. Insuch flows part of grains flows past each other while othergrains maintain static contacts with their neighbors. The

description of such flows still represents a major challengefor the theory. In particular, one is faced with the prob-lem of constructing the constitutive relation for the stresstensor σ. In dense quasi-static flows a significant part ofthe stresses is transmitted through quasi-static contactsbetween particles as compared with short collisions indilute flows.

Stimulated by the non-trivial avalanche dynamics inexperiments by Daerr (2001); Daerr and Douady (1999),Aranson and Tsimring (2001, 2002) suggested a genericcontinuum description of partially fluidized granularflows. According to this theory, the ratio of the staticpart σ

s to the fluid part σf of the full stress tensor is

controlled by the order parameter ρ. The order param-eter is scaled in such a way that in granular solid ρ = 1and in well developed flow (granular liquid) ρ → 0. Onthe “microscopic level” the order parameter is defined asa fraction of the number of static (or persistent) contactsof the particles Zs to total number of the contacts Z,ρ = 〈Zs/Z〉 within a mesoscopic volume which is largewith respect to the particle size but small compared withcharacteristic size of the flow.

Due to a strong dissipation in dense granular flows theorder parameter ρ is assumed to obey purely relaxationaldynamics controlled by the Ginzburg-Landau-type equa-tion for the generic first order phase transition,

Dρ

Dt= D∇2ρ− ∂F (ρ, δ)

∂ρ. (29)

Here D is the diffusion coefficient. F (ρ, δ) is a free energydensity which is postulated to have two local minima atρ = 1 (solid phase) and ρ = 0 (fluid phase) to accountfor the bistability near the solid-fluid transition.

The relative stability of the two phases is controlled bythe parameter δ which in turn is determined by the stresstensor. The simplest assumption consistent with theMohr-Coloumb yield criterion is to take it as a function ofφ = max |σmn/σnn|, where the maximum is sought overall possible orthogonal directions m and n (we considerhere only two-dimensional formulation of the model, anobjective three dimensional generalization was recentlyproposed by Gao et al. (2005)). Furthermore, there aretwo angles which characterize the fluidization transitionin the bulk of granular material, an internal friction an-gle tan−1 φ1 such that if φ > φ1 the static equilibrium isunstable, and the “dynamic repose angle” tan−1 φ0 suchthat at φ < φ0, the “dynamic” phase ρ = 0, is unstable.Values of φ0 and φ1 depend on microscopic propertiesof the granular material, and in general they do not co-incide. Aranson and Tsimring (2001, 2002) adopted thesimple algebraic form of the control parameter δ,

δ = (φ2 − φ20)/(φ2

1 − φ20). (30)

Order parameter equation (29) has to be augmented byboundary conditions. While this is a complicated issue ingeneral, a simple but meaningful choice is to take no-fluxboundary conditions at free surfaces and smooth walls,

17

FIG. 25 Comparison of theoretical and experimental phasediagrams. Lines obtained from theory, symbols depicts ex-perimental data from Ref. (Daerr and Douady, 1999). Solidline and circles limit the range of existence of avalanches,line and triangles correspond to the linear stability bound-ary of the static chute, and the line and crosses denote theboundary between triangular and balloon avalanches. In-set: Schematic representation of a chute flow geometry, fromAranson and Tsimring (2001, 2002)

FIG. 26 Sequence of images demonstrating the evolutionof a triangular avalanche (a-c) and up-hill avalanche (d-f)obtained form numerical solution of Eqs. (32),(31), fromAranson and Tsimring (2001, 2002)

and solid phase condition ρ = 1 near sticky or roughwalls.

For the flow of thin granular layers on inclined planesEqs. (6), (29) can be simplified. Using the no-slipboundary condition at the bottom and no-flux condi-tion at the top of the layer and fixing the lowest-modestructure of the order parameter in the direction per-pendicular to the bottom of the chute (z = 0, see Insetto Fig. 25), ρ = 1 − A(x, y) sin(πz/h), h is the locallayer thickness, A(x, y) is slowly-varying function, oneobtains equations governing the evolution of thin layer(Aranson and Tsimring, 2001, 2002):

∂th = −α∂x(h3A) +α

φ∇

(

h3A∇h)

, (31)

∂tA = λA+ ∇2⊥A+

8(2 − δ)

3πA2 − 3

4A3 (32)

where ∇2⊥ = ∂2

x + ∂2y , λ = δ − 1 − π2/4h2, α ≈

0.12µ−1g sin ϕ, µ is the shear viscosity, ϕ is the chuteinclination, φ = tan ϕ. Control parameter δ includescorrection due to change in the local slope δ = δ0 + βhx,β ≈ 1/(φ1 − φ0) ≈ 1.5 − 3 depending on the value ofφ. The last term in Eq. (31) is also due to change oflocal slope angle ϕ and is obtained from the expansionϕ ≈ ϕ + hx. This term is responsible for the saturationof the avalanche front slope (without it the front wouldbe arbitrarily steep). While it was not included in orig-inal publications (Aranson and Tsimring, 2001, 2002),this term is important for large wavenumber cut-off oflong-wave instability observed by Forterre and Pouliquen(2003), see Sec. VI.C. Numerical and analytic solutionsof Eqs. (31),(32) exhibit strong resemblance with exper-iment: triangular avalanches in thin layers and balloon-like avalanches in thicker layers, see Fig. 26. The cor-responding phase diagram agrees quantitatively with anexperimental one having only one fitting parameter (vis-cosity µ), Fig. 25.

In subsequent work (Aranson and Tsimring, 2002),this theory was generalized to other dense sheargranular flows including flows in rotating drums,two- and three-dimensional shear cells, etc. The

model also was tested in soft-particle moleculardynamics simulations (Volfson et al., 2003a,b, 2004).Orpe and Khakhar (2005) used the partial fluidiza-tion model of (Aranson and Tsimring, 2001, 2002;Volfson et al., 2003a) for the description of velocity pro-files three-dimensional shear flows in a rotating drum.The comparison between experimental data and theoryshows that the partial fluidization model describes rea-sonably well entire velocity profile and the flow rheology,however experimental methods for independently esti-mating the order parameter model are needed. Gao et al.

(2005) recently developed an objective (coordinate sys-tem independent) formulation of the partial fluidizationtheory which allows for the straightforward generaliza-tion to three-dimensional systems.

2. Two-phase flow approach of granular avalanches

Another approach treating near-surface granularflows as two-phase systems was developed by anumber of authors, see e.g. (Bouchaud et al.,1994, 1995; Boutreux et al., 1998; Douady et al., 1999;Khakhar et al., 2001; Mehta, 1994) and many oth-ers. For review on recent models of surface flows see(Aradian et al., 2002). All these models distinguishrolling and static phases of granular flow described bythe set of coupled equations for the evolution of thick-nesses of both phases, R and h, respectively. The phe-nomenological theory by Bouchaud et al. (1994, 1995);Mehta (1994) (often called BCRE theory) provides anintuitive description of the flow. In shallow granularlayers, even simpler depth-averaged granular hydrody-namic equations (so-called Saint-Venant models) oftenprovides quite accurate description, see (Douady et al.,1999; Khakhar et al., 2001; Lajeunesse et al., 2004;Savage and Hutter, 1989).

In the most general and compact form the BCRE the-ory can be represented by a pair of equations for evolutionof R and h,

∂th = Γ(h,R) (33)

∂tR = vd∂xR− Γ(h,R) (34)

where Γ(h,R) is the exchange term, or a conversion ratebetween rolling and static grains, and vd is the downhillgrain velocity. Physical meaning of the BCRE model isvery simple: Eq. (33) expresses the increase in the heightdue to deposition of rolling grains, and Eq. (34) describesadvection of rolling fraction by the flow with velocityvd and depletion due to conversion to static fraction.The limitations and generalizations of the BCRE modelare discussed by Aradian et al. (2002); Boutreux et al.

(1998).Douady et al. (2002) applied the following two-phase

model to describe avalanches in thin granular layers:

∂th+ 2U∂xh =g

Γ(tanφ− µ(h)) (35)

∂tζ + 2U∂xh = 0 (36)

18

where ζ = R+h is the position of free surface, U is depth-averaged velocity of the flow. In addition to BCRE modelEqs. (35),(36) include two phenomenological functions:Γ characterizes the mean velocity gradient of a singlebead on incline, and µ(h) describes depth-dependent fric-tion with the bottom. According to Douady et al. (2002)a three-dimensional version of Eqs. (35),(36) describestransition from triangular to uphill avalanches, howeverdetails of the transition depend sensitively on the choiceof functions Γ and µ(h).

Depth-averaged description in the form of Eqs.(35),(36) was used by Borzsonyi et al. (2005) to addressthe difference between shapes of avalanches for sand andglass particles in a chute flow. The authors reduced Eqs.(35),(36) to the modified Burgers equation

∂th+ a(h)∂xh = µ(h)∂2xh (37)

where function a(h) ∼ h3/2 and effective viscosity µ ∼√h. This description connects avalanches with the

“Burgers” shocks. Eq. (37) implies that all avalancheswill eventually decay, in contrast to experiments indi-cating that only small avalanches decay whereas largeavalanches grow and/or form stationary waves (Daerr,2001a; Daerr and Douady, 1999). This discrepancy islikely due to the fact that reduction of the full model(35),(36) to the single equation (37) does not take intoaccount the bistable nature of granular flows.

While two-phase description of granular flow is sim-ple and rather intuitive, it can be problematic when aclear-cut separation between rolling and static phases isabsent, especially near the onset of motion. The orderparameter approach can be more appropriate in this sit-uation. Furthermore, the two-phase equations can be de-rived from the partial fluidization model described in theSec. VI.A.1 as a sharp-interface limit of the continuumorder parameter model (Aranson and Tsimring, 2002).

3. Avalanche shape

On the basis of simple kinematic considerationsRajchenbach (2002b) suggested an analytic expressionfor the shape of triangular and balloon-like avalanches.For the balloon-like avalanches the shape is given by theenvelope of the expanding circles with the center driftingdownhill:

x2 +

(

y − 2vt+5

2vτ

)2

=

(

1

2vτ

)2

, 0 < τ < t (38)

where v is the velocity of the rear front. For the trian-gular avalanches the shape is given by the envelope ofdilating ellipses

(

vx

2v⊥

)2

+

(

y − 3

2vt

)2

=

(

1

2vt

)2

. (39)

Here v⊥ is perpendicular velocity. While these heuristicrelations are consistent with experimental observation,

FIG. 27 Top: Total area overrun by the avalanche (solid line),compared with experimental image from (Daerr and Douady,1999). Bottom: superimposition of avalanche boundariesgiven by Eq. (39) for three different moments of time, fromRajchenbach (2002b)

see Fig. 27, their connection to continuum dynamicalmodels of granular flows remains to be understood.

B. Statistics of avalanches and sandpile model

It is well known that in real sandpiles avalanches canvary widely in size. The wide distribution of scales inreal avalanches stimulated Bak et al. (1987) to introducea “sandpile cellular automaton” as a paradigm model forthe self-organized criticality, the phenomenon which oc-curs in slowly driven non-equilibrium spatially extendedsystems when they asymptotically reach a critical statecharacterized by a power-law distribution of event sizes.The set of rules which constitute the sandpile modelis very simple. Unit size “grains” are dropped one byone on a one-dimensional lattice in random places andform vertical stacks. If a local slope (the difference be-tween heights of two neighboring stacks) exceeds a cer-tain threshold value, a grain hops from the higher to thelower stack. This may trigger an “avalanche” of subse-quent hops until the sandpile returns to the stable state.After that another grain is dropped and the relaxationprocess repeats. The size of an avalanche is determinedby the number of grains set into motion by adding a singlegrain to a sandpile. This model asymptotically reachesa critical state in which the mean angle is equal to thecritical slope, and avalanches have a universal power-lawdistribution of sizes, P (s) ∝ s−α with α ≈ 1.5.

The relevance of this model and its generalizations tothe real avalanches is still the matter of debate. Thesandpile model by Bak et al. (1987) is defined via a sin-gle repose angle and so its asymptotic behavior has theproperties of the critical state for a second-order phasetransition. Real sandpiles are characterized by two anglesof repose and thus exhibit features of the first-order phasetransition. Moreover, concept of self-organized criticalityis related to a power-law distribution of avalanche sizes,thus reliable experimental verification of self-organizedcriticality requires accumulation of very large statisticsof avalanche events and a large-scale experimental setup.Finite size effects should strongly affect the power-lawbehavior.

Experiments with avalanches in slowly rotating drums(Jaeger et al., 1989; Rajchenbach, 2000) and chute flows(Lemieux and Durian, 2000) do not confirm the scale-invariant distribution of avalanches. In other experi-ments with large mono-disperse glass beads dropped on aconical sandpile Costello et al. (2003) claimed existenceof the self-organized criticality with α ≈ 1.5. Character-istics of the size distribution depended on the geometry

19

of the sandpile, physical and geometrical properties ofgrains, and the way the grains are dropped on the pile,contrary to the universal concept of self-organized criti-cal behavior. Self-organized criticality was also claimedin the avalanche statistics in three-dimensional pile ofanisotropic grains (long rice), however a smaller scalingexponent α ≈ 1.2 was measured for the avalanche sizedistribution (Aegerter et al., 2004). Interestingly, ricepiles were observed to demonstrate roughening dynam-ics of their surface as the distribution of active sites inthe self-organized critical state shows a self-affine struc-ture with the fractal exponent dB = 1.85 (Aegerter et al.,2004). This is consistent with the theoretically predictedmapping between self-organized criticality and roughen-ing observed for example in Kardar-Parizi-Zhang model(Paczuski and Boettcher, 1996).

One can argue that real sandpiles should not exhibitself-organized criticality in a strict sense due to hysteresisand the existence of two critical repose angles. However,since the difference between the angles is relatively small,one cannot exclude power-law type behavior in the fi-

nite range of avalanche sizes. This circumstance possiblyexplains significant scatter in experimental results andscaling exponents for avalanche size distribution and thedependence on grain shape and material properties.

C. Instabilities in granular chute flows

Granular chute flows exhibit a variety of pattern-forming instabilities, including fingering (Malloggi et al.,2005a; Pouliquen et al., 1997), longitudinal vortices(Borzsonyi and Ecke, 2005; Forterre and Pouliquen,2001, 2002), long surface waves (Forterre and Pouliquen,2003), segregation and stratification (Gray and Hutter,1997a; Makse et al., 1997b), etc.

Pouliquen et al. (1997) studied experimentally a gran-ular chute flow on a rough inclined plane. Experimentsperformed with polydisperse sand particles demonstratedfingering instability of the front propagating down theslope, similar to that observed in fluid films flowing downinclined plane (Troian et al., 1989; Zhou et al., 2005).However, similar experiments with smooth monodisperseglass beads exhibited no instability. The authors arguedthat the instability was due to a flow-induced size segre-gation in a polydisperse granular matter. The segrega-tion indeed was found near the avalanche front. However,similar experiments (Malloggi et al., 2005a,b) showed afingering front instability without a significant size segre-gation. Thus, the question of the mechanism of fingeringinstability is still open.

Experiments by Borzsonyi and Ecke (2005);Forterre and Pouliquen (2001, 2002) show the de-velopment of longitudinal vortices in rapid chute flows,see Fig. 8. The vortices develop for large inclination an-gles and large flow rates in the regime of accelerating flowwhen the flow thickness decreases and the mean flow ve-locity increases along the chute. Forterre and Pouliquen

FIG. 28 Density profiles ν(z) as function of distance form thechute bottom z for different values of mean flow velocity, fromForterre and Pouliquen (2002)

FIG. 29 Phase diagram in mean density (ν) and flow thick-ness (h) plane delineating different flow instabilities. Smallernu corresponds to faster flow, from Forterre and Pouliquen(2002)

(2001) proposed an explanation of this phenomenon interms of granular “thermoconvection”. Namely, rapidgranular flow has a high shear near the rough bottomwhich leads to the local increase of granular temperatureand consequently creates a density inversion. In turn,the density inversion trigger a convection instabilitysimilar to that in ordinary fluids. The critical instabilitywavelength λC is determined by the depth of the layer h(in experiment λc ≈ 3h).

In a subsequent work Forterre and Pouliquen (2002)studied the formation of longitudinal vortices and thestability of granular chute flows in the framework of gran-ular hydrodynamics Eqs.(5)-(7). The inverse density pro-file appears when a heuristic boundary condition at thebottom relating slip velocity and heat flux is introduced.Steady-state solution of Eqs.(5)-(7) indeed yields an in-verse density profile (Fig. 28) which turns out to beunstable with respect to short-wavelength perturbationsfor large flow velocities, see Fig. 29. While the linear sta-bility analysis captured many important features of thephenomenon, there are still open questions. The stabil-ity analysis was performed for the steady flow whereasthe instability occurs in the regime of accelerating flow.Possibly due to this assumption the linear stability anal-ysis yielded oscillatory instability near the onset of vor-tices, whereas for the most part, the vortices appear tobe steady. Another factor which is ignored in the theoryis the air drag. The high flow velocity in the experiment(about 1-2 m/sec) is of the order of the terminal velocityof an individual grain in air, and therefore air drag mayaffect the granular flow.

Forterre and Pouliquen (2003) presented an experi-mental study of the long-surface-wave instability devel-oping in granular flows on a rough inclined plane, Fig.30. This instability was known from previous stud-ies (Davies, 1990; Savage, 1979), however no precisecharacterization of the instability had been performed.Forterre and Pouliquen (2003) measured the thresholdand the dispersion relation of the instability by impos-ing a controlled perturbation at the entrance of the flowand measuring its evolution down the slope, see Fig. 31.The results are compared with the prediction of a linear

FIG. 30 Long-surface wave instability observed in flow of sanddown rough incline, from Forterre and Pouliquen (2003)

20

FIG. 31 Experimental dispersion relation for the long surfacewave instability. Shown spatial growth rate as function ofthe frequency of forcing wave, from Forterre and Pouliquen(2003)

FIG. 32 The growth rate of small perturbations σ vs wave-length k derived from Eqs. (31),(32) for β = 2, α =0.025, δ = 1.1. Instability occurs near hstop curve in Figs.24,25 (h = 2.9) and disappears with further increase of h.

stability analysis conducted in the framework of depth-averaged Saint-Venant-type equations similar to thosedescribed in Sec. VI.A.2:

∂th + ∂x(uh) = 0 (40)

∂t(uh) + α∂x(u2h) = (tan θ − µ(u, h) − ∂xh ) gh cos(θ)

where h is local thickness, θ is inclination angle, uis depth-averaged flow velocity, µ(h, u) is a functiondescribing effective depth and velocity dependent bot-tom friction, α ∼ O(1) is a constant determined bythe velocity profile within the layer. According toForterre and Pouliquen (2003), the instability is similarto the long-wave instability observed in classical fluidsbut with characteristics that can dramatically differ dueto the specificity of the granular rheology. The theory isable to predict quantitatively the stability threshold andthe phase velocity of the waves but fails to describe theobserved cutoff of the instability at high wavenumbers.Most likely, one needs to include higher order terms, suchas ∂2

xh in the first Eq. (40) in order to account for thecutoff.

The order parameter theory based on Eqs. (31),(32)also reproduces the long-surface wave instability. Fur-thermore, linearizing Eqs. (31),(32) near the steady flow-

ing solution A = A0 + a exp[σt+ ikx], h = h0 + h exp[σt+ikx], after simple algebra one obtains that the growthrate of linear perturbations σ is positive only in a bandrestricted by some critical wavenumber and only in thevicinity of hstop, see Fig. 32. With the increase of h, i.e.the granular flux, the instability disappears, in agree-ment with experiments. The nonlinear saturation of theinstability results in the development of a sequence ofavalanches, which is generally non-periodic, see Fig. 33.The structure shows slow coarsening due to merging ofthe avalanches. This instability is a possible candidatemechanism of the formation of inhomogeneous depositstructure behind the front of an avalanche.

Conway et al. (2003) studied free-surface waves ingranular chute flows near a frictional boundary. The

FIG. 33 Typical profiles of hight h and order parameter Ain the regime of long-surface wave instability for β = 2,α = 0.025, δ = 1.1. Starting from generic initial conditionsh = h0, A = const plus small noise, a sequence of avalanchesdevelops.

experiments showed that the sub-boundary circulationdriven by the velocity gradient plays an important rolein the pattern formation, suggesting a similarity betweenwave generation in granular and fluid flows.

A Kelvin-Helmholtz-like shear instability in chuteflows was observed by Goldfarbs et al. (2002), when twostreams of sand flowing on an inclined plane with dif-ferent velocities were in side-by-side contact with eachother. For sufficiently high chute angles and shearrates the interface remains flat. The instability of theinterface develops when the chute angle and/or theshear rate is reduced. This instability has been repro-duced in soft-particle molecular dynamics simulations byCiamarra et al. (2005) who also observed that in a poly-disperse medium this instability leads to grain segrega-tion (See below Sect. VII).

D. Pattern-forming instabilities in rotating cylinders

Granular media in rotating horizontal cylinders(drums) often show behavior similar to chute flows. Forvery small rotation rates (as defined by small Froudenumber Fr = ω2R/g, where ω is the angular velocity ofdrum rotation and R its radius), well separated in timeavalanches occur when the slope of the free surface ex-ceeds a certain critical angle θc whereby diminishing thisangle to a smaller static repose angle θs (Jaeger et al.,1989; Rajchenbach, 1990; Tegzes et al., 2002, 2003). Thedifference between θc and θs is usually a few degrees. Atan intermediate rotation speed, a continuous flow of sandemerges instead of discrete avalanches through a hys-teretic transition, similar to the transition in chute flowsat large rates of grain deposition (Lemieux and Durian,2000). In the bulk, the granular material rotates almostas a solid body with some internal slipping. As mov-ing grains reach the free surface they slide down withina thin near-surface layer (Zik et al., 1994) (see sketch inFig. 9). The surface has a nearly flat shape; the arct-angent of its average slope defines the so-called dynamic

angle of repose θd.There are various models addressing the nature of

the transition from discrete avalanches to the continuumflow. Linz and Hanggi (1995) proposed a phenomenolog-ical model based on a system of equation for the angle ofrepose φ and mean flow velocity v

v = g (sinφ− k(v) cosφ))χ(φ, v)

φ = ω − av (41)

where ω is the rotation frequency of the drum, k(v) =b0 + b2v

2 is the velocity dependent friction coefficient,χ(φ, v) is some cut-off function, and a, b0, b2 are parame-ters of the model. Despite the simplicity, the model yieldsqualitatively correct transition from discrete avalanchesto continuous flow with the increase of rotation rate ω,and also predicts logarithmic relaxation of the free sur-face angle in the presence of vibration.

21

The transition from avalanches to flow naturally arisesin the framework of the partial fluidization theory,(Aranson and Tsimring, 2002). In this case one can de-rive a system of coupled equations for the parameter δ(which is related to the surface local angle φ, see Eq.(30)) and the width of fluidized layer z0,

∂tz0 = ∂2sz0 + F (z0, δ) − v∂sz0

∂tδ = ω + ∂2sJ (42)

where s is the coordinate along the slope of the granularsurface inside the drum, J = f(z0) is the downhill flux ofgrains, v is averaged velocity in flowing layer, and func-tions F, f and v0 are derived from Eq. (29). This modelbears resemblance to the BCRE-type models of surfacegranular flows which were applied to rotating drums byKhakhar et al. (1997); Makse (1999).

Eqs. (42) exhibit stick-slip type oscillations of the sur-face angle for slow rotation rates and a hysteretic transi-tion to a steady flow for larger rates. Eqs. (42) yields thefollowing scaling for the width of the flowing layer z0 inthe middle of the drum vs rotation frequency: z0 ∼ ω2/3,which is consistent with experiment (Tegzes et al., 2002,2003). After integration over s Eqs. (42) can be reducedto a system of two coupled equations for averaged drumangle 〈δ〉 and averaged flow thickness 〈z0〉 somewhat sim-ilar to the model of Linz and Hanggi (1995).

Granular flows in long rotating drums under certainconditions also exhibit fingering instability (Fried et al.,1998; Shen, 2002). Similarity between fingering in ro-tating drums and chute flows (Forterre and Pouliquen,2002) suggests that mechanisms described in the SectionVI.C can be responsible for this effect, see also SectionVII.

VII. MODELS OF GRANULAR SEGREGATION

One of the most fascinating features of heterogeneous(i.e., consisting of different distinct components) granu-lar materials is their tendency to segregate under externalagitation rather than to mix, as one would expect fromthe naive entropy consideration. This property is ubiq-uitous in Nature (see, e.g. (Iverson, 1997)) and has im-portant technological implications (Cooke et al., 1976).In fact, some aspects of segregation of small and largeparticles can be understood on equilibrium thermody-namics grounds (Asakura and Oosawa, 1958). Since theexcluded volume for small particles around large onesbecomes smaller when large grains clump together, sep-arated state possesses lower entropy. However, granu-lar systems are driven and strongly dissipative, and thissimple equilibrium argument can only be applied qual-itatively. The granular segregation is more widespreadthan it would be dictated by thermodynamics. In fact,any variation in mechanical properties of particles (size,shape, density, surface roughness, etc.) may lead to theirsegregation. At least for bi-disperse rapid dilute flowsthe granular segregation can be rigorously treated in the

FIG. 34 Granular stratification in a flow down heap, fromMakse et al. (1997b).

framework of kinetic theory of dissipative gases, see Sub-sec. III.A. Jenkins and Yoon (2002) employed kinetictheory for a binary mixture for spheres or disks in grav-ity and derived a simple segregation criterion based onthe difference of partial pressures for each type of parti-cles due to the difference in size and/or mass.

Segregation has been observed in most flows ofgranular mixtures, including granular convection(Knight et al., 1993), hopper flows (Gray and Hutter,1997a; Makse et al., 1997b; Samadani et al., 1999;Samadani and Kudrolli, 2001), flows in rotatingdrums (Choo et al., 1997; Hill, 1997; Zik et al., 1994),and even in freely cooling binary granular gases(Catuto and Marconi, 2004). Segregation among largeand small particles due to shaking has been termed“Brazil nut effect” (Rosato et al., 1987). The phe-nomenon of granular segregation was discovered longtime ago, and several “microscopic” mechanisms havebeen proposed to explain its nature, including inter-particle collisions (Brown, 1939), percolation (Williams,1976), and others. In certain cases, separation of grainsproduces interesting patterns. For example, if a binarymixture of particles which differ both in size and inshape is poured down on a plane, a heap which consistsof thin alternating layers of separated particles is formed(Gray and Hutter, 1997a; Makse et al., 1997b), see Fig.34. Rotating of mixtures of grains with different sizes inlong drums produces well separated bands of pure mono-disperse particles (Chicarro et al., 1997; Choo et al.,1997; Hill, 1997; Zik et al., 1994), Fig. 11. In thisSection we only address models of pattern formationdue to segregation (stratification and banding), withoutdiscussing other manifestations of granular segregation.

A. Granular stratification

Granular stratification occurs when a binary mix-ture of particles with different physical properties isslowly poured on a plate (Gray and Hutter, 1997a;Koeppe et al., 1998; Makse et al., 1997b). More specifi-cally, it occurs when larger grains have additionally largerroughness resulting in a larger repose angle and the fluxof falling particles is small enough to cause intermittentavalanches down the slopes of the heap. The basic mecha-nism of stratification is related to the avalanches acting askinetic sieves (Savage, 1988, 1993). During an avalanche,voids are continuously being created within flowing near-surface layer, and small particles are more likely to fallinto them. This creates a downward flux of smaller par-ticles which is compensated by the upward flux of largerparticles in order to maintain a zero total particle fluxacross the flowing layer. Other models of granular seg-regation in a thin flowing layer (Dolgunin et al., 1998;

22

FIG. 35 Cellular automata model of granular stratification,from Makse et al. (1997a).

Khakhar et al., 1997, 1999) lead to a similar result. Eachavalanche leads to the formation of a new pair of layersin which the grains of different sorts are separated (seeFig. 34). This pair of layers grows from the bottom ofthe pile by upward propagation of a kink at which smallparticles are stopped underneath large ones. However,when the larger particles were smooth and small parti-cles were rough, instead of stratification only large scalesegregation with small particles near the top and largeparticles near the bottom was observed.

Makse et al. (1997a,b) proposed a cellular automatamodel which generalized the classical sandpile model(Bak et al., 1987) (see Section VI.B). In this model, asandpile is built on a lattice, and rectangular grain haveidentical horizontal size but different heights (see Fig.35a). Grains are released at the top of the heap sequen-tially, and they are allowed to roll down the slope. Aparticle would become rolling if the local slope (definedas the height difference between neighboring columns)exceeds the repose angle. To account for difference ingrain properties, four different repose angles θαβ were in-troduced for grains of type α rolling on a substrate oftype β (α, β ∈ 1, 2 where 1 and 2 stand for small andlarge grains, respectively). Normally, θ21 < θ12 becauseof the geometry (small grains tend to get trapped bylarge grains), and one-component repose angles usuallylie within this range, θ21 < θ11, θ22 < θ12. However theratio of θ11, θ22 depends on the relative roughness of thegrains. For θ21 < θ11 < θ22 < θ12 (large grains are morerough), the model yields stratification in agreement withexperiment (Fig. 35b). If, on the other hand, θ22 < θ11(which corresponds to smaller grains being more rough),the model yields only large-scale segregation: large par-ticles collect at the bottom of the sandpile.

This physical model can also be recast in the formof continuum equations (Boutreux and de Gennes, 1996;Makse et al., 1997a) which generalize the single-speciesBCRE model of surface granular flows (Bouchaud et al.,1994) (see Section VI):

∂tRα = −vα∂xRα + Γα, (43)

∂th = −∑

α

Γα, (44)

whereRα(x, t), vα are the thickness and velocity of rollinggrains of type α, h(x, t) is the instantaneous profile ofthe sandpile, and Γα characterizes interaction betweenthe rolling grains and the substrate of static grains. Inthe same spirit as in the discrete model, the interactionfunction Γα is chosen in the form

Γα =

γα[θl − θα(φβ)]Rα

γαφα[θl − θα(φβ)]Rα. (45)

Here φα(x, t) is the volume fraction of grains of type α,and θl = −∂xh is the local slope of the sandpile. This

FIG. 36 Granular avalanche-induced stratification in rotatingdrum observed for low rotation rates, from Gray and Hutter(1997a).

form of the interaction terms implies that the grains oftype α become rolling if the local slope exceeds the reposeangle θα(φβ) for this type on a surface with compositionφβ(x, t). Assuming that the generalized repose anglesθα(φβ) are linear functions of the concentration

θ1(φ2) = (θ12 − θ11)φ2 + θ11, (46)

θ2(φ2) = (θ12 − θ11)φ2 + θ21. (47)

Eqs. (43)-(45) possess a stationary solution in whichthe heap is separated into two regions where θ2(φ2) <θ < θ1(φ2) and θ < θ2(φ2) < θ1(φ2). This solutioncorresponds to small grains localized near the top andsmall grains near the bottom with a continuous tran-sition between the two regions. However, Makse et al.

(1997b) showed that this stationary solution is unstableif δ = θ22 − θ11 > 0 and gives rise to the stratificationpattern.

Similar effect of stratification patterns was observedexperimentally in a thin slowly rotating drum whichis more than half filled with a similar binary mix-ture (Gray and Hutter, 1997a), see Fig. 36. Periodicavalanches, occurring in the drum, lead to formation ofstrata by the same mechanism described above.

B. Axial segregation in rotating drums

The most common system in which granular segrega-tion is studied is a rotating drum, or a partially filledcylinder rotating around its horizontal axis (see SectionVI.D). When a polydisperse mixture of grains is ro-tated in a drum, strong radial segregation usually occurswithin just a few revolutions. Small and rough parti-cles aggregate to the center (core) of the drum, large andsmooth particles rotate around the core (see Figs. 10and 9). Since there is almost no shear flow in the bulk,the segregation predominantly occurs within a thin flu-idized near-surface layer. For long narrow drums withthe length much exceeding the radius, radial segregationis often followed by the axial segregation occurring atlater stages (after several hundred revolutions) when theangle of repose of small particles exceeds that of largeparticles. As a result of axial segregation, a pattern ofwell segregated bands is formed (Hill, 1997; Zik et al.,1994) (see, e.g., Fig. 11) which slowly merge and coarsen.Depending on the rotation speed, coarsening can eithersaturate at a certain finite bandwidth at low rotationspeeds when discrete avalanches provide granular trans-port (Frette and Stavans, 1997) or at higher rotationrates in a continuous flow regime it can lead to a fi-nal state in which all sand is separated in two bands(Fiodor and Ottino, 2003; Zik et al., 1994).

23

The axial segregation has been well known in the en-gineering community, it was apparently first observedby Oyama in 1939 (Oyama, 1939). The mechanism ofaxial segregation is apparently related to the differentfriction properties of grains which lead to different dy-

namical angles of repose. The latter are defined as theangle of the slope in the drum corresponding to contin-uous flow regime, however in real drums the free surfacehas a more complicated S-shape (Elperin and Vikhansky,1998; Makse, 1999; Orpe and Khakhar, 2001; Zik et al.,1994). According to Zik et al. (1994) (see also (Levine,1999)), if there is a local increase in concentration of par-ticles with higher dynamic repose angle, the local slopethere will be higher, and that will lead to a local bumpnear the top of the free surface and a dip near the bot-tom. As the particles tend to slide along the steepestdescent path, more particles with higher repose anglewill accumulate in this location, and the instability willdevelop. Zik et al. (1994) proposed a quantitative con-tinuum model of axial segregation based on the equa-tion for the conservation equation for the relative con-centration of the two components (“glass” and “sand”),c(z, t) = (ρA − ρB)/(ρA + ρB),

∂tc = − C

ρT(tan θA−tan θB)∂z(1−c2)〈(1+y2

x)yz

yx〉. (48)

Here x and z are Cartesian horizontal coordinates acrossand along the axis of the drum, y(x, z, t) describes the in-stantaneous free surface inside the drum, ρT = ρA + ρB,C is a constant related to gravity and effective viscosityof granular material in the flowing layer. The term inangular brackets denotes the axial flux of the glass beadsaveraged over the cross-section of the drum. The pro-file of the free surface in turn should depend on c(z, t).If 〈(1 + y2

x)yc/yx〉 < 0, linearization of Eq.(48) leads tothe diffusion equation with negative diffusion coefficientwhich exhibits segregation instability with growth rateproportional to the square of the wavenumber. It is easyto see that the term in angular brackets vanishes for astraight profile yx = const(x). However, for the exper-imentally observed S-shaped profile of the free surfaceZik et al. (1994) calculated that the instability conditionis satisfied when the drum is more than half full. Whileexperiments show that axial segregation in fact observedeven for less than 50% filling ratio, the model gives a goodintuitive picture for the mechanism of the instability.

Recent experiments (Choo et al., 1997, 1998;Fiodor and Ottino, 2003; Hill, 1997) have revealedinteresting new features of axial segregation. Hill (1997)performed magnetic resonance imaging studies (Hill,1997) which demonstrated that in fact the bands oflarger particles usually have a core of smaller particles.More recent experiments by Fiodor and Ottino (2003)showed that small particles formed a shish kebab-likestructure with bands connected by a rod-like core, whilelarge particles formed disconnected rings. Choo et al.

(1997, 1998) found that at early stages, the small-scaleperturbations propagate across the drum in both direc-

FIG. 37 Space-time diagram of the surface of long rotatingdrum demonstrating oscillatory size segregation. The plotshows the full length of the drum and extends over 2,400 sec,or 1,850 revolutions. Black bands correspond to 45-250 µmblack sand and white bands correspond to 300-850 µm tablesalt, from Choo et al. (1997).

tions (this was clearly evidenced by the experiments onthe dynamics of pre-segregated mixtures (Choo et al.,1997)), while at later times more long-scale staticperturbations take over and lead to the emergence ofquasi-stationary bands of separated grains (see Fig. 37).The slow coarsening process can be accelerated in adrum of a helical shape (Zik et al., 1994). Alternatively,the bands can be locked in an axisymmetrical drum withthe radius modulated along the axis (Zik et al., 1994).

In order to account for the oscillatory behavior of ax-ial segregation at the initial stage, Aranson and Tsimring(1999); Aranson et al. (1999b) generalized the model ofZik et al. (1994). The key assumption was that be-sides the concentration difference, there is an addi-tional slow variable which is involved in the dynamics.Aranson and Tsimring (1999); Aranson et al. (1999b)conjectured that this variable is the instantaneous slopeof the granular material (dynamic angle of repose) whichunlike Eq.(48) is not slaved to the relative concentrationc, but obeys its own dynamics. The equations of themodel read

∂tc = −∂z(−D∂zc+ g(c)∂zθ), (49)

∂tθ = α(Ω − θ + f(c)) +Dθ∂zzθ + γ∂zzc. (50)

The first term in the r.h.s. of Eq.(49) describes diffusionflux (mixing), and the second term describes differentialflux of particles due to the gradient of the dynamic reposeangle. This term is equivalent to the r.h.s. of Eq.(48)with a particular function g(c) = G0(1 − c2). For sim-plicity, the constant G0 can be eliminated by rescaling ofdistance x→ x/

√G0. The sign + before this term means

that the particles with the larger static repose angle aredriven towards greater dynamic repose angle. This differ-ential flux gives rise to the segregation instability. Sincethis segregation flux vanishes with g(c) |c| → 1 (whichcorrespond to pure A or B states), it provides a naturalsaturation mechanism for the segregation instability.

Parameter Ω in the second equation is the normalizedangular velocity of the drum rotation, and f(c) is thestatic angle of repose which is an increasing function ofthe relative concentration (Koeppe et al., 1998) (for sim-plicity it can be assumed linear, f(c) = F+f0c). The con-stant F can be eliminated by the substitution θ → θ−F .The first term in the r.h.s. of Eq.(50) describes the lo-cal dynamics of the repose angle (Ω increases the angle,and −θ + f(c) describes the equilibrating effect of thesurface flow), and the term Dθ∂xxθ describes axial dif-fusive relaxation. The last term, γ∂xxc, represents the

24

FIG. 38 Dispersion relation λ(k) for segregation instability(left) and comparison of the frequency of band oscillationsIm(λ) with experiment (right), from Aranson and Tsimring(1999).

FIG. 39 Space-time diagrams demonstrating initialband oscillations and consequent coarsening, fromAranson and Tsimring (1999).

lowest-order non-local contribution from an inhomoge-neous distribution of c (the first derivative ∂xc cannotbe present due to reflection symmetry x → −x). Thisterm gives rise to the transient oscillatory dynamics ofthe binary mixture.

Linear stability analysis of a homogeneous state c =c0; θ0 = Ω + f0c0 reveals that for g0f0 > αD long-waveperturbations are unstable, and if g0γ > (Dθ − D)2/4,short-wave perturbations oscillate and decay (two eigen-values λ1,2 are complex conjugate with negative realpart), see Fig. 38. This agrees with the general phe-nomenology observed by Choo et al. (1997) both quali-tatively and even quantitatively (Fig. 38b). The resultsof direct numerical solution of the full model (49),(50)are illustrated by Fig. 39. It shows that short-waveinitial perturbations decay and give rise to more long-wave non-oscillatory modulation of concentration whicheventually leads to well-separated bands. At long times(Fig. 39b) bands exhibit slow coarsening with thenumber of bands decreasing logarithmically with time(see also Fiodor and Ottino (2003); Frette and Stavans(1997); Levitan (1998)). This scaling follows from theexponentially weak interaction between interfaces sep-arating different bands (Aranson and Tsimring, 1999;Fraerman et al., 1997).

While these continuum models of axial segregationshowed a good qualitative agreement with the data, re-cent experimental observations demonstrate that the the-oretical understanding of axial segregation is far fromcomplete (Ottino and Khakhar, 2000). The interpreta-tion of the second slow variable as the local dynamicangle of repose implies that in the unstable mode theslope and concentration modulation should be in phase,whereas in the decaying oscillatory mode, these twofields have to be shifted in phase. Further experiments(Khan et al., 2004) showed that while the in-phase rela-tionship in the asymptotic regime holds true, the quadra-ture phase shift in the transient oscillatory regime is notobserved. That lead Khan et al. (2004) to hypothesizethat some other slow variable other than the angle ofrepose (possibly related to the core dynamics) may beinvolved in the transient dynamics. However, so far ex-periments failed to identify which second dynamical fieldis necessary for oscillatory transient dynamics, so it re-mains an open problem. Another recent experimentalobservation by Khan and Morris (2005) suggested thatinstead of normal diffusion assumed in Eqs.(49),(50), a

slower subdiffusion of particles in the core takes place,〈r〉 ∼ tγ with the scaling exponent γ close to 0.3. Themost plausible explanation is that the apparent subd-iffusive behavior is in fact a manifestation of nonlinear

concentration diffusion which can be described by equa-tion

∂tc = ∂zD(c)∂zc. (51)

For example, for the generic concentration-dependent dif-fusion coefficient D ∼ c, the asymptotic scaling behaviorof the concentration c(z, t) is given by the self-similarfunction c ∼ F (z/tα)/tα for t → ∞ with the scaling ex-ponent α = 1/3 close to 0.3 observed experimentally. Ex-perimentally observed scaling function F (x/tα) appearsto be consistent with that of Eq. (51) except for thetails of the distribution where c→ 0 and the assumptionD ∼ c is possibly violated. Normal diffusion behaviorcorresponding to D = const and α = 1/2 is in strongdisagreement with the experiment.

Newey et al. (2004) conducted studies of axial segre-gation in ternary mixtures of granular materials. It wasfound that for certain conditions bands of ternary mix-tures oscillate axially. In contrast to the experiments ofChoo et al. (1997, 1998), the oscillations of bands appearspontaneously from initially mixed state, which stronglyindicates the supercritical oscillatory instability. Whilein binary mixtures the oscillations have the form of peri-odic mixing/demixing of bands, in the ternary mixturesthe oscillations are in the form of periodic band displace-ments. It is likely that the mechanism of band oscillationsin ternary mixtures is very different from that of binarymixtures. One of possible explanations could be that thethird mixture component provides an additional degree offreedom necessary for oscillations. To demonstrate thatwe write phenomenological equations for the concentra-tion differences CA = c1 − c2 and CB = c2 − c3, wherec1,2,3 are the individual concentrations. By analogy withEq. (49) we write the system of coupled equations for theconcentration differences CA,B linearized near the fullymixed state:

∂tCA = DA∂2zCA + µA∂

2zCB ,

∂tCB = DB∂2zCB + µB∂

2zCA. (52)

If the cross-diffusion terms have opposite signs, i.e.µAµB < 0, the concentrations CA,B will exhibit oscil-lations in time and in space. Obviously this mechanismis intrinsic to ternary systems and has no counterpart inbinary mixtures.

Parallel to the theoretical studies, molecular dynam-ics simulations have been performed (Rapaport, 2002;Shoichi, 1998; Taberlet et al., 2004). Simulations al-lowed researchers to probe the role of material param-eters which would be difficult to access in laboratoryexperiments. In particular, Rapaport (2002) addressedthe role of particle-particle and wall-particle friction co-efficients separately. It was found that the main role

25

FIG. 40 Granular Taylor vortices observed in vertically ro-tating air-fluidized cylinder filled with binary mixture. Leftimage depicts entire cylinder height and width, and right im-age shows the dependence of concentration of small particlesalong the bed height, from Shinbrot (2004).

is played by the friction coefficients between the parti-cles and the cylinder walls: if the friction coefficient be-tween large particles and the wall is greater than thatfor smaller particles, the axial segregation always oc-cur irrespective of the ratio of particle-particle frictioncoefficients. However, if the particle-wall coefficientsare equal, the segregation may still occur if the frictionamong large particles is greater than among small parti-cles. Taberlet et al. (2004) studied axial segregation in asystem of grains made of identical material differing onlyby size. The simulations revealed rapid oscillatory mo-tion of bands, which is not necessarily related to the slowband appearence/disappearence observed in experimentsof Choo et al. (1997, 1998); Fiodor and Ottino (2003).

A different type of discrete element modelling of ax-ial segregation was proposed by Yanagita (1999). Thismodel builds upon the lattice-based sandpile model andreplaces a rotating drum by a three-dimensional squarelattice. Drum rotation is modelled by correlated displace-ment of particles on the lattice: particles in the back areshifted upward by one position, and the particles at thebottom are shifted to fill the voids. This displacementsteepens the slope of the free surface, and once it reachesa critical value, particles slide down according to the rulessimilar to the sandpile model of Bak et al. (1987) buttaking into account different critical slopes for differentparticles. This model despite its simplicity reproducedboth radial and axial segregation patterns and thereforeelucidated the critical components needed for adequatedescription of the phenomenon.

C. Other examples of granular segregation

As we have seen in the previous Section, granular seg-regation occurs in near-surface shear granular flows, suchas in silos, hoppers, and rotating drums. However, othertypes of shear granular flows may also lead to segrega-tion. For example, Taylor-Couette flow of granular mix-tures between two rotating cylinders leads to formation ofTaylor vortices and then in turn to segregation patterns(Shinbrot, 2004), see Fig. 40.

Pouliquen et al. (1997) observed granular segregationin a thin granular flow on an inclined plane. In this case,segregation apparently occurs as a result of an instabilityin which concentration mode is coupled with hydrody-namic mode. As a result, segregation occurs simultane-ously with a fingering instability of the avalanche front(Fig. 7). As an implicit evidence of this relation betweensegregation and fingering instability, Pouliquen et al.

(1997) found that mono-disperse granular material does

not exhibit fingering instability. However, other exper-iments (Shen, 2002) indicate that in other conditions(more rapid flows), fingering instability may occur evenin flows of mono-disperse granular materials. Thus, thesegregation is likely a consequence rather than the pri-mary cause of the fingering instability.

An interesting recent example of pattern formationcaused by granular segregation in a horizontally shakenlayer of binary granular mixture was presented by Mullin(2000, 2002); Reis and Mullin (2002). After several min-utes of horizontal shaking with frequency 12.5 Hz anddisplacement amplitude 1 mm (which corresponds to theacceleration amplitude normalized by gravity Γ = 0.66),stripes were formed orthogonal to the direction of shak-ing. The width of the stripes was growing continuouslywith time as d ∝ t0.25, thus indicating slow coarsening(Fig. 5). This power law is consistent with the diffusion-mediated mechanism of stripe merging. Reis and Mullin(2002) argued on the basis of experimental results on pat-terned segregation in horizontally shaken layers that thesegregation bears features of the second-order phase tran-sition. Critical slow-down was observed near the onset ofsegregation. The order parameter is associated with thecombined filling fraction C, or the layer compacity,

C =NsAs +NlAl

S(53)

where Ns,l are numbers of particles in each species, As,l

are projected two-dimensional areas of the respective in-dividual particles, and S is the tray area. Ehrhard et al.

(2005) proposed a simple numerical model to describethis phenomenon of segregation in horizontally vibratedlayers. The model is based on a two-dimensional systemof hard disks of mass mα and radius Rα (α = 1, 2 denotethe species)

mαvαi = −γi (vαi − vtray(t)) + ζαi(t) (54)

where vi is the particles velocity vtray(t) = A0 sin(ωt)is oscillating tray velocity, γ provides linear damping,and ζαi is Gaussian white noise acting independently oneach disk. The model reproduced segregation instabil-ity and subsequent coarsening of stripes. More realis-tic discrete element simulations were recently performedby Ciamarra et al. (2005). In these simulations a binarymixture of round disks of identical sizes but two differ-ent frictions with the bottom plate (in fact, velocity-dependent viscous drag was assumed), separated in al-ternating bands perpendicular to the oscillation direc-tion irrespectively on initial conditions: both randommixed state and separated along the direction of oscil-lations state were used. Using particles of the same sizeeliminated the thermodynamic “excluded volume” mech-anism for segregation, and the authors argued that themechanism at work is related to the dynamical shear in-stability similar to the Kelvin-Helmholtz instability inordinary fluids. It was confirmed by a numerical obser-vation of the interfacial instability when two monolay-ers of grains with different friction constant were placed

26

in contact along a flat interface parallel to the direc-tion of horizontal oscillations. Similar instability is ap-parently responsible for ripple formation (Scherer et al.,1999; Stegner and Wesfreid , 1999).

Pooley and Yeomans (2004) proposed theoretical de-scription of this experiment based on continuum modelfor periodically-driven two isothermal ideal gases whichinteract through frictional force. It was shown analyt-ically that segregated stripes form spontaneously abovecritical forcing amplitude. While the model reproducesthe segregation instability, apparently it does not exhibitcoarsening of stripes observed in the experiment. More-over, applicability of the isothermal ideal gas model tothis experiment where the particles are almost at rest isan open question.

Similar coarsening effect in granular segregationin a particularly simple geometry was studied byAumaitre et al. (2001). They investigated the dynamicsof a monolayer of grains of two different sizes in a dishshaken in a horizontal “swirling” motion. They observedthat large particles tend to aggregate near the center ofthe cavity surrounded by small particles. The qualitativeexplanation of this effect follows from simple thermody-namic considerations (see above). Indeed, direct tracingof particle motion showed that the pressure in the areanear the large particles is smaller than outside. But smallparticles do not follow the gradient of pressure and as-semble near the center of the cavity because this gradientis counterbalanced by the force from large particles. Theinverse of force acting on large particles leads to their ag-gregation near the center of the cavity. Aumaitre et al.

(2001) proposed a more quantitative model of segregationbased on the kinetic gas theory and found satisfactoryagreement with experimental data.

Burtally et al. (2002) studied spontaneous separationof vertically vibrated mixtures of particles of similar sizesbut different densities (bronze and glass spheres). Atlow frequencies and at sufficient vibrational amplitudes,a sharp boundary between the lower layer of glass beadsand the upper layer of the heavier bronze spheres wasobserved. At higher frequencies, the bronze particlesemerge as a middle layer separating upper and lower glassbead layers. The authors argue that the effect of air onthe granular motion is a relevant mechanism of particleseparation. A somewhat similar conclusion was achievedby Mobius et al. (2001) in experiments with vertically-vibrated column of grains containing a large “intruder”particle.

Arndt et al. (2005); Fiodor and Ottino (2003) per-formed detailed experiments on axial segregation in slur-ries, or bi-disperse grain-water mixtures. A mixture oftwo types of spherical glass beads of two sizes were placedin a water-filled tube at the volume ratio 1:2. Authorsfound that both rotation rate and filling fraction play animportant role in band formation. Namely, bands areless likely to form at lower fill levels (20-30%) and slowerrotation rates (5-10 rpm). They mostly appear near theends of the drum. At higher fill levels and rotation rates,

bands form faster, and there are more of them through-out the drum. Arndt et al. (2005); Fiodor and Ottino(2003) also studied the relation between the bands visi-ble on the surface, and the core of small beads, and foundthat for certain fill levels and rotation speeds, the core re-mains prominent at all times, while in other cases the coredisappears completely between bands of small particles.They also observed an interesting oscillatory instabilityof interfaces between bands at high rotation speeds. Allthese phenomena still await theoretical modelling.

VIII. GRANULAR MATERIALS WITH COMPLEX

INTERACTIONS

A. Patterns in solid-fluid mixtures

Presence of interstitial fluid significantly complicatesthe dynamics of granular materials. Hydrodynamic flowslead to the viscous drag and anisotropic long-range inter-action between particles. Even small amounts of liquidleads to cohesion among the particles which can havea profound effect on macroscopic properties of granu-lar assemblies such as angle of repose, avalanching, abil-ity to segregate, etc. (see for example Sec. VII.B andLi and McCarthy (2005); Samadani and Kudrolli (2000,2001); Tegzes et al. (2002)).

In this Section we will discuss the case when the vol-ume fraction of fluid in the two-phase system is large,and the grains are completely immersed in fluid. Thisis relevant for many industrial applications, as well asfor geophysical problems such as sedimentation, erosion,dune migration, etc.

One of the most technologically important examplesof particle-laden flows is a fluidized bed. Fluidized bedshave been widely used since German engineer Fritz Win-kler invented the first fluidized bed for coal gasifica-tion in 1921. Typically, a vertical column containinggranular matter is energized by a flow of gas or liquid.Fluidization occurs when the drag force exerted by thefluid on the granulate exceeds gravity. A uniform flu-idization, the most desirable regime for most industrialapplications, turns out to be prone to bubbling insta-bility: bubbles of clear fluid are created spontaneouslyat the bottom, traverse the granular layer and destroythe uniform state (Jackson, 2000). Instabilities in flu-idized beds is an active area of research in the engi-neering community, see (Gidaspow, 1994; Jackson, 2000;Kunii and Levenspiel, 1991). A shallow fluidized bedshows many similarities with mechanically vibrated lay-ers, see Section V. In particular, modulations of airflowstudied by Li et al. (2003) result in formation of subhar-monic square and stripe patterns (see Fig. 23) similarto those in mechanically-vibrated systems (Melo et al.,1994, 1995; Umbanhowar et al., 1996).

Wind and water driven granular flows play importantroles in geophysical processes. Wind-blown sand formsdunes and beaches. The first systematic study of air-

27

borne (or aeolian) sand transport was conducted by R.Bagnold during Wold War II, see Bagnold (1954) whoidentified two primary mechanisms of sand transport:saltation and creep, and proposed the first empiric re-lation for the sand flux q driven by wind shear stress τ :

q = CBνa

g

√

d

Du3∗ (55)

where CB = const, νa is air density, d is the grain di-ameter, D = 0.25 mm is a reference grain size, andu∗ =

√

τ/νa is wind friction velocity. Later many refine-ments of Eq.(55) were proposed, see e.g. (Pye and Tsoar,1991).

Nishimori and Ouchi (1993) proposed a simple theorywhich describes formation of ripples as well as dunes.The theory is based on a lattice model which incorpo-rates separately saltation and creep processes. The modeloperates with the height of sand at each lattice side atdiscrete time n, hn(x, y). The full time step includes twosubsteps. The saltation substep is described as

hn(x, y) = hn(x, y) − q (56)

hn(x+ L(h(x, y)), y) = hn(x+ L(hn(x, y)), y) + q

where q is the height of grains being transferred fromone (coarse grained) position (x, y) to the other position(x+L, y) on the lee side (wind is assumed blowing in thepositive x direction), L is the flight length in one saltationwhich characterizes the wind strength. It is assumed thatq is conserved. Since the saltation length L depends onmultiple factors, the following simple approximation isaccepted

L = L0 + bhn(x, y) (57)

with L0 measuring wind velocity and b = const. Thecreep substep involves spatial averaging over neighboringsites in order to describe the surface relaxation due togravity,

hn+1(x, y) = hn(x, y) + (58)

D

[

1

6

∑

NN

h(x, y) +1

12

∑

NNN

h(x, y) − h(x, y)

]

,

where∑

NN and∑

NNN denote summation over thenearest neighbors and next nearest neighbors corre-spondingly, and D = const is the surface relaxationrate. Despite its simplicity, simulation of the modelreproduced formation of ripples and consequently ar-rays of barchan (crescent shaped) dunes, see Fig. 41.Nishimori and Ouchi (1993) found that above certainthreshold an almost linear relation holds between theselected wavelength of the dune pattern and the “windstrength” L.

In the long-wave limit Eqs. (57)-(59) can be reducedto more traditional continuum models considered below.

FIG. 41 Sand ripple pattern (top panel) and barchan dunes(lower panel) obtained from simulations of Eqs. (57)-(59)Nishimori and Ouchi (1993)

FIG. 42 Interaction and coarsening of one-dimensional dunesystem, from Prigozhin (1999)

In the continuum description of the evolution of thesand surface, the profile h is governed by the mass con-servation equation

νs∂th = −∇q, (59)

where νs is the density of sand and q is the sand flux.In order to close Eqs. (55),(59), several authors pro-posed different phenomenological relations between shearstress at the bed surface τ and the height h, see e.g.(Andreotti et al., 2002; Hersen et al., 2004; Kroy et al.,2002a,b; Nishimori and Ouchi, 1993; Prigozhin, 1999).

There are many theories generalizingNishimori and Ouchi (1993) approach, see e.g.(Caps and Vaanderwalle, 2001). Prigozhin (1999)described the evolution of dunes by a system of twoequations similar to the BCRE model discussed earlierin Sec. VI (Bouchaud et al., 1994, 1995). One equationdescribes the evolution of the local height h whileanother equation describes the density R of particlesrolling above the stationary sand bed profile (reptatingparticles),

∂th = Γ(h,R) − f (60)

∂tR = −∇J + Q− Γ(h,R) (61)

where Γ is the rolling-to-steady sand transition rate, J

is the horizontal projection of the flux of rolling parti-cles, Q accounts for the influx of falling reptating grains,and f is the erosion rate. With an appropriate choiceof rate functions Γ, f,Q and J , Eqs. (61) can reproducemany observed features of dune formation, such as initialinstability of flat state, asymmetry of the dune profiles,coarsening and interaction of dunes, etc., see Fig. 42.

Thus, simplified models such as (Kroy et al., 2002a;Nishimori and Ouchi, 1993; Prigozhin, 1999) have beensuccessful in explaining many features of individual dunegrowth and evolution, see Fig. 43. However we shouldnote that up to date none of the dune models have beenable to address satisfactorily the wavelength selection inlarge-scale dune fields (Hersen et al., 2004).

The phenomenon qualitatively similar to the dune for-mation occurs in an oscillatory fluid flow above a gran-ular layer: sufficiently strong flow oscillations produce

FIG. 43 Evolution of two barchan dunes described by Eqs.(55), (59). Small dune is undersupplied and eventuallyshrinks, from Hersen et al. (2004)

28

FIG. 44 Fingering instability of planar avalanche in the un-derwater flow, figure on the left zooms on individual finger,from Malloggi et al. (2005a)

so-called vortex ripples on the surface of the underlyinggranular layer. These ripples are familiar to any beach-goer. Vortex ripple formation was first studied by Ayrton(1910); Bagnold (1956), and recently by Scherer et al.

(1999); Stegner and Wesfreid (1999) and others. It wasfound that ripples emerge via a hysteretic transition, andare characterized by a near-triangular shape with slopeangles close to the repose angle. The characteristic sizeof the ripples λ is directly proportional to the displace-ment amplitude of the fluid flow a (with a proportionalityconstant ≈ 1.3) and is roughly independent on the fre-quency.

Andersen et al. (2001) introduced order parametermodels for describing the dynamics of sand ripple pat-terns under oscillatory flow based on the phenomenolog-ical mass transport law between adjacent ripples. Themodels predict the existence of a stable band of wavenumbers limited by secondary instabilities and coarsen-ing of small ripples, in agreement with experimental ob-servations.

Langlois and Valance (2005) studied underwater rip-ple formation on a two-dimensional sand bed sheared byviscous fluid. The sand transport is described by gener-alization of Eq. (55) taking into account both the localbed shear stress and the local bed slope. Linear stabil-ity analysis revealed that ripple formation is attributedto a growing longitudinal mode. The weakly nonlin-ear analysis taking into account resonance interaction ofonly three unstable modes revealed a variety of steadytwo-dimensional ripple patterns drifting along the flowat some speed.

Experiments in dune formation have been recently per-formed in water (Betat et al., 1999). While water-drivenand wind driven dunes and ripples have similar shape,the underlying physical processes are likely not the samedue to a different balance between gravity and viscousdrag in air and water.

Spectacular erosion patterns in sediment granular lay-ers were observed in experiments with underwater flows(Daerr et al., 2003; Malloggi et al., 2005a). In particu-lar, a fingering instability of flat avalanche fronts wasobserved, see Fig. 44. These patterns are remarkablysimilar to those in thin films on inclined surfaces, bothwith clear and particle-laden fluids (Troian et al., 1989;Zhou et al., 2005). In the framework of lubrication ap-proximation the evolution of fluid film thickness h is de-scribed by the following dimensionless equation followingfrom the mass conservation law:

∂th+ ∇ ·[

h3∇∇2h]

− Dh3∇h

+ ∂xh3 = 0 (62)

where dimensionless parameter D is inversely propor-tional to water surface tension. The instability occurs

FIG. 45 Upper panel: Self-supporting knolls formed in wa-ter/glass beads suspension in a horizontally rotating cylinder(side and end views). Lower panel: Computational results:schematics of the flow (a); the height of computed knoll struc-tures (b,c), from Duong et al. (2004)

for small D values, i.e. in the large surface tension limit.However, despite visual similarity the physical mecha-nism leading to this fingering instability is not obvious:in fluid films the instability is driven (and stabilized) bythe surface tension, whereas in the underwater granularflow fluid surface tension plays no role.

Duong et al. (2004) studied formation of periodic ar-rays of knolls in a slowly rotating horizontal cylinder filledwith granular suspension, see Fig. 45. The solidified sed-iment knolls co-exist with freely circulating fluid. Theauthors applied variable viscosity fluid which formallyallows simultaneous treatment of solid and liquid phase.In this model the effective flow viscosity µs diverges atthe solid packing fraction φrcp,

µs =µ0

(1 − φ/φrcp)b(63)

where µ0 is the clear fluid viscosity and b is an empir-ical coefficient. The model qualitatively reproduces theexperiment, see Fig. 45. An interesting question in thiscontext is whether there is a connection to the experi-ment by Shen (2002) where somewhat similar structureswere obtained for the flow of “dry” particles in a hori-zontally rotating cylinder.

As it was mentioned in Sec. VII.C, Conway et al.

(2004) reported that an air-fluidized vertical columnof bi-disperse granular media sheared between counter-rotating cylinders exhibits formation of nontrivial vortexstructure strongly reminiscent of Taylor vortices in con-ventional fluid, see e.g. (Andereck et al., 1986). Authorsargue that vortices in fluidized granular media, unlikeTaylor vortices in fluid, are accompanied by the novelsegregation-mixing mechanism specific for granular sys-tems, see Fig. 40. Interestingly, no vortices were ob-served in a similar experiment in Couette geometry withmonodisperse glass beads (Losert et al., 2000).

Ivanova et al. (1996) studied patterns in a horizontalcylinder filled with sand/liquid mixture and subject tohorizontal vibration. For certain vibration parametersstanding wave patterns were observed at the sand/liquidinterface. Authors argue that these wave patterns aresimilar to the Faraday ripples found at liquid/liquid in-terface under vertical vibration.

B. Vortices in vibrated rods

In Section V we reviewed instabilities and collectivemotion in mechanically vibrated layers. In most exper-iments the particle shape was not important. However,strong particles anisotropy may give rise to non-trivial

29

FIG. 46 Phase diagram for the system of vertically-vibratedrods, driving frequency 50 Hz. Vortices are observed for suffi-ciently high filling fraction nf and above critical accelerationΓ, from Blair et al. (2003a)

FIG. 47 Azimuthal velocity of the vortex vs distance from thecenter for different parameter values, from Blair et al. (2003a)

effects. Villarruel et al. (2000) observed onset of nematicorder in packing of long rods in a narrow vertical tubesubjected to vertical tapping. The rods initially com-pactify into a disordered state with predominantly hori-zontal orientation, but at later times (after thousands oftaps) they align vertically, first along the walls, and thenthroughout the volume of the pipe. The nematic order-ing can be understood in terms of the excluded volumeargument put forward by Onsager (1949).

Blair et al. (2003a) studied the dynamics of vibratedrods in a shallow large aspect ratio system. Surprisingly,they found that vertical alignment of rods at large enoughfilling fraction nf and the amplitude of vertical accelera-tion (Γ > 2.2) can occur in the bulk, and it does not re-quire side walls. Eventually, most of the rods align them-selves vertically in a monolayer synchronously jumpingon the plate, and engage in a correlated horizontal mo-tion in the form of propagating domains of tilted rods,multiple rotating vortices etc, see Fig. 12 and Fig. 46.The vortices exhibit almost rigid body rotation near thecore, and then the azimuthal velocity falls off, Fig. 47.The vortices merge in the course of their motion, andeventually a single vortex is formed in the cell.

Experiments showed that the rod motion occurs whenthe rods are tilted from the vertical, and it alwaysoccurs in the direction of tilt. In subsequent workVolfson et al. (2004) experimentally demonstrated thatthe correlated transport of bouncing rods is also foundin quasi-one-dimensional geometry, and explained thiseffect using molecular dynamics simulations and a de-tailed description of inelastic frictional contacts be-tween the rods and the vibrated plate. Effectively,bouncing rods become self-propelled objects similarto other self-propelled systems, for which large-scalecoherent motion is often observed (bird flocks, fishschools, chemotactic microorganism aggregation, etc., seee.g. Gregoire and Chate (2004); Helbing et al. (2000);Helbing (2001); Toner and Tu (1995)).

Aranson and Tsimring (2003) developed a phe-nomenological continuum theory describing coarseningand vortex formation in the ensemble of interactingrods. Assuming that the motion of rods is overdampeddue to the bottom friction, the local horizontal velocityv = (vx, vy) of rods is of the form

v = − (∇p− αnf0(n)ν) /ζν, (64)

where ν is the density, p is the hydrodynamic pressure,the tilt vector n = (nx, ny) is the projection of the rod

FIG. 48 Sequence of images illustrating coalescence of vor-tices in the model of vibrated rods, the field |n| is shown,black dots corresponds to vortex cores where |n| = 0, fromAranson and Tsimring (2003)

director on the (x, y) plane normalized by the rod length,i.e n = |n|, and ζ is friction coefficient. According toBlair et al. (2003a); Volfson et al. (2004), the rods driftis determined by the average tilt of neighboring rods,thus the term αnf0(n)ν accounts for the average drivingforce from the vibrating bottom on the tilted rod. Eq.(64) combined with the mass conservation law yields

∂tν = −div(vν) = ζ−1div (∇p− αnf0(n)ν) . (65)

To account for the experimentally observed phase sepa-ration and coarsening Aranson and Tsimring (2003) em-ployed the Cahn-Hilliard approach (see (Bray, 1994) forreview) by assuming that pressure p can be obtained fromthe variation of a generic bistable “free energy” func-tional F with respect to the density field ν, p = δF/δν.

To close the description the equation for the evolutionof tilt n is added on generic symmetry arguments:

∂tn = f1(ν)n − |n|2n +

+ f2(ν)(

ξ1∇2n + ξ2∇divn)

+ β∇ν. (66)

Here f1,2 are certain functions of ν, ξ1,2 characterize dif-fusion coupling between the neighboring rods. Since thetilt field is not divergence-free, from the general symme-try considerations both ξ1,2 6= 0 3.

Numerical and analytic studies of Eqs. (65),(66) re-vealed phase coexistence, nucleation and coalescence ofvortices in accord with the experiment, see Fig. 48.

An interesting experiment with anisotropic chiral par-ticles was performed by Tsai et al. (2005). The role ofparticles was played by bend-wire objects which rotatedin a preferred direction under vertical vibration. The ex-periments demonstrated that individual angular rotationof the particles was converted into the collective angularmomentum of the granular gas of these chiral objects.The theoretical description of this system was formulatedin the framework of two phenomenological equations forthe density ν and center-of-mass momentum density νvand the spin angular momentum density l = IΩ aris-ing from the ration of particles around their center ofmass, Ω is the particle’s rotation frequency. Whereas theequations for density and velocity are somewhat similarto those for the vibrated rod system, the equation forthe spin momentum clearly has no counterpart in the vi-brated rod system and was postulated in the followingform:

∂tl + ∇vl = τ − ΓΩ − Γ(Ω − ω) +DΩ∇2Ω (67)

3 These constants are analogous to the first and second viscosityin ordinary fluids, see e.g. (Landau and Lifshits, 1959)

30

where τ is the source of the angular rotation (due to chi-rality of particles), ω is coarse-grained or collective an-gular velocity, ΓΩ and Γ are dissipative coefficients dueto friction and DΩ is the angular momentum diffusion.Eq. (67) predicts, in agreement with the experiment, theonset of collective rotation of the gas of particles. Possi-bly, it also exhibits non-trivial spatio-temporal dynamicssimilar to those in the system of vibrated rods. How-ever, due to the small number of particles (about 350) inthe experiment the nontrivial collective regimes were notreported.

C. Electrostatically driven granular media

Large ensembles of small particles display fascinatingcollective behavior when they acquire an electric chargeand respond to competing long-range electromagneticand short-range contact forces. Many industrial tech-nologies face the challenge of assembling and separatingsuch single- or multi-component micro and nano-size en-sembles. Traditional methods, such as mechanical vibra-tion and shear, are infective for very fine powders due toagglomeration, charging, etc. Electrostatic effects oftenchange statistical properties of granular matter such asenergy dissipation rate (Sheffler and Wolf, 2002), veloc-ity distributions in granular gases (Aranson and Olafsen,2002; Kohlstedt et al., 2005), agglomeration rates in sus-pensions (Dammer and Wolf, 2004), etc.

Aranson et al. (2000, 2002); Sapozhnikov et al.

(2003a, 2004) studied electrostatically driven granularmatter. This method relies on the collective interactionsbetween particles due to a competition between shortrange collisions and long-range electromagnetic forces.Direct electrostatic excitation of small particles offersunique new opportunities compared to traditionaltechniques of mechanical excitation. It enables one todeal with extremely fine nonmagnetic and magneticpowders which are not easily controlled by other means.

In most experimental realizations, several grams ofmono-dispersed conducting micro-particles were placedinto a 1.5 mm gap between two horizontal 30 × 30cm2 glass plates covered by transparent conducting lay-ers of indium tin-dioxide. Typically 45 µm Copperor 120 µm Bronze spheres were used. Experimentswere also performed with much smaller 1 µm particles,(Sapozhnikov et al., 2004). An electric field perpendicu-lar to the plates was created by a high voltage source (0-3kV) connected to the inner surface of each plate. Experi-ments were performed in air, vacuum, or in the cell filledwith non-polar weakly-conducting liquid.

The basic principle of the electro-cell operation is asfollows. A particle acquires an electric charge when itis in contact with the bottom conducting plate. It thenexperiences a force from the electric field between theplates. If the upward force induced by the electric fieldexceeds gravity, the particle travels to the upper plate,reverses charge upon contact, and is repelled down to the

bottom plate. This process repeats in a cyclical fashion.In an air-filled or evacuated cell, the particle remains im-mobile at the bottom plate if the electric field E is smallerthan the first critical field E1. For E > E1 an isolatedparticle leaves the plate and starts to bounce. However,if several particles are in contact on the plate, screeningof the electric field reduces the force on individual par-ticles, and they remain immobile. A simple calculationshows that for the same value of the applied electric fieldthe force acting on isolated particles exceeds by a factorof two the force acting on the particle inside the densemonolayer. However, if the field is larger than a secondcritical field value, E2 > E1, all particles leave the plate,and the system of particles transforms into an uniformgas-like phase. When the field is decreased below E2

(E1 < E < E2), in air-filled or evacuated cells localizedclusters of immobile particles spontaneously nucleate toform a static clusters (precipitate) on the bottom plate(Aranson et al., 2000). The clusters exhibit the Ostwald-type ripening (Bray, 1994; Meerson, 1996), see also Sub-sec. IV.C.

1. Coarsening of clusters

Results for the electrostatically driven system yieldedthe following asymptotic scaling law, see Fig. 50:

N ∼ 1

t(68)

where N is the number of clusters and t is time. Accord-ingly, the average cluster area 〈A〉 increases with time as〈A〉 ∼ t. This behavior is consistent with the interface-

controlled Ostwald ripening (Meerson, 1996).A theoretical description of coarsening in an elec-

trostatically driven granular system was developed byAranson et al. (2000), Sapozhnikov et al. (2003). Thetheory was formulated in terms of the Ginzburg-Landau-type equation for the number density of immobile parti-cles (precipitate or solid) n

∂tn = ∇2n+ φ(n, ng) (69)

where ng is the number density of bouncing parti-cles (gas) ng, and φ(n, ng) is a function characterizinga solid/gas conversion rate. The effectiveness of thesolid/gas transitions is controlled by the local gas con-centration ng. It was assumed that the gas concentrationis almost constant because the particle’s mean free passin the gas state is very large. The gas concentration ng

is coupled to n due to total mass conservation constraint

Sng +

∫ ∫

n(x, y)dxdy = M, (70)

where S is the area of domain of integration, and Mis the total number of particles. Function φ(n, ng) ischosen in such a way as to provide bistable local dynamicsof concentration corresponding to the hysteresis of the

31

FIG. 49 Illustration of phase separation and coarsening dy-namics. (a)-(c) Numerical solution of Eqs. (69), (70), whitecorresponds to dense clusters, black to dilute gas. (d)-(f) showexperimental results, from Aranson et al. (2002).

gas/solid transition. The above description yields a verysimilar temporal evolution of clusters (see Fig. 49) andproduces a correct scaling for the number of clusters Eq.(68).

In the so-called sharp interface limit when the size ofclusters is much larger than the width of interfaces be-tween clusters and granular gas, Eq. (69) can be reducedto equations for the cluster radii Ri (assuming that clus-ters have circular form):

dRi

dt= κ

(

1

Rc(t)− 1

Ri

)

, (71)

where Rc is critical cluster size, κ is effective surface ten-sion (experimental measurements of cluster surface ten-sion were conducted by Sapozhnikov et al. (2003)). Thecritical radius Rc is a certain function of the granular gasconcentration ng that enters Eqs. (71) through the theconservation law Eq. (70) which in two dimensions reads

ngS + π

N∑

i=1

R2i = M. (72)

The statistical properties of Ostwald ripening canbe understood in terms of the probability distri-bution function f(R, t) of cluster sizes. FollowingLifshitz and Slyozov (1958, 1961); Wagner (1961) and

neglecting cluster merger, one obtains in the limit N →∞ that the probability distribution f(R, t) satisfies thecontinuity equation

∂tf + ∂R

(

Rf)

= 0. (73)

From the mass conservation in the limit of small gas con-centration Eq. (72) one obtains an additional constraint:

π

∫ ∞

0

R2f(R, t)dR = M (74)

Eqs. (73),(74) have a self-similar solution in the form

f(R, t) =1

t3/2F

(

R√t

)

(75)

For the total number of clusters N =∫ ∞

0fdR the scal-

ing Eq.(75) yields N ∼ 1/t, which appears to be in agood agreement with the experiment, see Fig. 50. How-ever, the cluster size distribution function appears to bein a strong disagreement, see Fig. 51. In particular,Lifshitz and Slyozov (1958, 1961); Wagner (1961) the-

ory predicts the distribution with a cut-off (dotted line)whereas the experiment yields the function with an ex-ponential tail. A much better agreement with the exper-iment was obtained when binary coalescence of clusters

FIG. 50 Average cluster area 〈A(t)〉 (a) and inverse num-ber of clusters 1/N(t) vs time in air-filled cell. The straightline in (b) shows theoretical prediction 1/N ∼ t, fromSapozhnikov et al. (2005)

FIG. 51 Scaled cluster size distribution function F (ξ) withξ = R/

√t. The squares show experimental results, the dot-

ted line shows analytic result form Lifshitz-Slyozov-Wagnertheory (Wagner, 1961), and solid line shows F obtainedfrom the theory accounting for binary coalescence, fromSapozhnikov et al. (2005)

was incorporated in the Lifshitz-Slyozov-Wagner theory(Conti et al., 2002; Sapozhnikov et al., 2005).The coales-cence events become important for a finite area fractionof the clusters.

Ben-Naim and Krapivsky (2003) applied an exchangegrowth model to describe coarsening in granular media.In this theory the cluster growth rates are controlled onlyby the cluster area ignoring shape effects. Assuming thatthe number of particles in a cluster evolves via uncorre-lated exchange of single particles with an other clusterthe following equation for the density of clusters contain-ing k particles can be derived:

dAk

dt=

∑

i,j

AiAjKij (δk,i+1 + δk,i−1 − 2δk,i) (76)

where Ak is the probability to find a cluster containingk particles, Kij the exchange kernel and δk,i is the Kro-necker symbol. For the choice of homogeneous kernelKij = (ij)λ with λ = 1 this theory predicts correct scal-

ing of the cluster size with time R ∼√t and exponential

decay of the cluster size distribution function, as in theexperiment. The choice of λ = 1 is equivalent to theassumption that the exchange rate is determined by thesize of the cluster. In the theory by Sapozhnikov et al.

(2005) the cluster evolution is governed by the evapora-tion/deposition of particles at the interface of the clusterand controlled by the overall pressure of the granular gas.Thus, both theories predict the same scaling behavior,however the underlying assumptions are very different.A possible explanation for this may be that while the ex-change growth model ignores the curvature of the clusterinterface and the dependence on exchange rate on thepressure of granular gas, the agreement is obtained bytuning the adjustable parameter λ.

2. Dynamics of patterns in a fluid-filled cell

Sapozhnikov et al. (2003a) performed experimentswith electrostatically driven granular media immersedin a weakly conducting non-polar fluid (toluene-ethanolmixture). Depending on the applied electric field andthe ethanol concentration (which controls the conductiv-ity of the fluid), a plethora of static and dynamic patterns

32

were discovered, see Fig. 15. For relatively low concen-trations of ethanol (below 3%), the qualitative behaviorof the liquid-filled cell is not very different from that ofthe air-filled cell: clustering of immobile particles andcoarsening were observed between two critical field val-ues E1,2 with the clusters being qualitatively similar tothat of the air cell. However, when the ethanol concen-tration is increased, the phase diagram becomes asym-metric with respect to the direction of the electric field.Critical field magnitudes, E1,2, are larger when the elec-tric field is directed downward (“+” on the upper plate)and smaller when the field is directed upward (“−” onthe upper plate). This difference increases with ethanolconcentration. The observed asymmetry of the criticalfields is apparently due to an excess negative charge inthe bulk of the liquid.

The situation changes dramatically for higher ethanolconcentrations: increasing the applied voltage leads tothe formation of two new immobile phases: honeycomb(Fig. 15b) for the downward direction of the appliedelectric field, and two-dimensional crystal-type states forthe upward direction.

A further increase of ethanol concentration leads to theappearance of a novel dynamic phase - condensate (Fig.15c,d) where almost all particles are engaged in a circularvortex motion in the vertical plane, resembling Rayleigh-Benard convection. The condensate co-exists with the di-lute granular gas. The direction of rotation is determinedby the polarity of the applied voltage: particles streamtowards the center of the condensate near the top platefor the upward field direction and vice versa. The evo-lution of the condensate depends on the electric field di-rection. For the downward field, large structures becomeunstable due to the spontaneous formation of voids (Fig.15d). These voids exhibit complex intermittent dynam-ics. In contrast, for the upward field, large vortices mergeinto one, forming an asymmetric object which often per-forms composite rotation in the horizontal plane. Thepattern formation in this system is most likely causedby self-induced electro-hydrodynamic micro-vortices cre-ated by the particles in weakly-conducting fluids. Thesemicro-vortices create long-range hydrodynamic vortexflows which often overwhelm electrostatic repulsion be-tween likely-charged particles and introduce attractivedipole-like hydrodynamic interactions. Somewhat simi-lar micro-vortices are known in driven colloidal systems,see e.g. (Yeh et al., 1997).

Aranson and Sapozhnikov (2004) developed a phe-nomenological continuum theory of pattern formation formetallic micro-particles in a weakly conducting liquidsubject to an electric field. Based on the analogy with thepreviously developed theory of coarsening in air-field cell(Aranson et al., 2002), the model is formulated in termsof conservation laws for the number densities of immobileparticles (precipitate) np and bouncing particles (gas) ng

averaged over the thickness of the cell:

∂tnp = ∇Jp + f , ∂tng = ∇Jg − f. (77)

FIG. 52 Sequence of snapshots illustrating evolution of pul-sating rings (top raw) and rotating vortices (bottom raw)obtained from numerical solution of Eqs. (77),(79), fromAranson and Sapozhnikov (2004)

Here Jp,g are the mass fluxes of precipitate and gas re-spectively and the function f describes gas/precipitateconversion which depends on np,g, electric field E andlocal ionic concentration c. The fluxes are written as:

Jp,g = Dp,g∇np,g + αp,g(E)v⊥np,g(1 − β(E)np,g), (78)

where v⊥ is horizontal hydrodynamic velocity, Dp,g areprecipitate/gas diffusivities. The last term, describ-ing particles advection by fluid, is reminiscent of theRichardson-Zaki relation for a drag force frequentlyused in the engineering literature (Richardson and Zaki,1954). The factor (1 − β(E)np,g) describes the satura-tion of flux at large particle densities n ∼ 1/β due to thedecrease of void fraction. Terms ∼ αp,g describe advec-tion of particles by the fluid. Interestingly, in the limit ofvery large gas diffusion Dg ≫ Dp and without advectionterms (αp,g = 0) the model reduces to Eqs. (69) and (70)applied for air-filled cell (Aranson et al., 2002).

Eqs. (77) are coupled to the cross section averagedNavier-Stokes equation for vertical velocity vz :

n0(∂tvz + v∇vz) = µ∇2vz − ∂zp+ Ezq (79)

where n0 is the density of liquid (we set n0 = 1), µ isthe viscosity, p is the pressure, and q is the charge den-sity. The last term describes the electric force acting oncharged liquid. Horizontal velocity v⊥ is obtained fromvz using the incompressibility condition ∂zvz +∇⊥v⊥ = 0in the approximation that vertical vorticity Ωz = ∂xvy −∂yvx is small compared to in-plane vorticity. This as-sumption allows one to find the horizontal velocity as agradient of quasi-potential φ: v⊥ = −∇⊥φ.

For an appropriate choice of the parameters the modelEqs. (77),(79) yields qualitatively correct phase diagramand the patterns observed in the experiment, see Figs.15 and 52.

D. Magnetic particles

Electric and magnetic interactions allow introduc-tion of controlled long-range forces in granular sys-tems. Blair et al. (2003a); Blair and Kudrolli (2003b);Stambaugh et al. (2004a,b) performed experimentalstudies with vibrofluidized magnetic particles. Severalinteresting phase transitions were reported, in particu-lar, the formation of dense two-dimensional clusters andloose quasi-one-dimensional chains and rings. Blair et al.

(2003a) considered pattern formation in a mixture ofmagnetic and non-magnetic (glass) particles of equalmass. The glass particles played the role of “phonons”,their concentration allowed an adjustment of the typi-cal fluctuation velocity of the magnetic subsystem. The

33

FIG. 53 Phase diagram illustrating various regimes in mag-netic granular media, T is the temperature determined fromthe the width of velocity distribution, Φ is surface coveragefraction of glass particles, from Blair and Kudrolli (2003b)

phase diagram delineating various regimes in this systemis shown in Fig. 53. While the phase diagram showssome similarity with equilibrium dipolar fluids (such asphase coexistence), most likely there are differences dueto the non-equilibrium character of granular systems.

Stambaugh et al. (2004a) performed experiments withrelatively large particles (about 1.7 cm), and near theclosed-packed density. It was found that particles formhexagonal closed-packed clusters in which the magneticdipoles lay in the plane and assume circulating vorti-cal patterns. For lower density ring patterns were ob-served. Experiments with mixture of particles with twodifferent magnetic moments revealed segregation effects(Stambaugh et al., 2004b). The authors argue that thestatic configurational magnetic energy is the primary fac-tor in pattern selection.

Experiments by Blair and Kudrolli (2003b);Stambaugh et al. (2004a,b) were limited to a smallnumber (about 103) of large particles due to the intrinsiclimitation of the mechanical vibrofluidization technique.Snezhko et al. (2005) performed experimental studies of90 µm Nickel micro-particles subjected to electrostaticexcitation, see also Subsec. VIII.C. The electrostaticsystem allowed researchers to perform experiments witha very large number of particles (of the order of 106)and a large aspect ratio of the experimental cell. Thusthe transition between small chains and large networks(Fig. 13) was addressed in detail. An abrupt divergenceof the chain length was found when the frequency offield oscillations decreased, resulting in the formation ofa giant interconnected network.

Studies of the collective dynamics and pattern forma-tion of magnetic particles are still in the early phases.While it is natural to assume that magnetic interactionplays a dominant role in pattern selection, further com-putational and theoretical studies of pattern formationin systems of driven dipolar particles are necessary. Be-sides a direct relevance for the physics of granular media,studies of magnetic granular media may provide an ad-ditional insight into the behavior of dipolar hard spherefluids where the nature of solid/liquid transitions is stilldebated (de Gennes and Pincus, 1970; Levin, 1999). Vi-bration or electrostatically fluidized magnetic particlescan also be viewed as a macroscopic model of a ferrofluid,where similar experiments are technically difficult to per-form.

IX. OVERVIEW AND PERSPECTIVES

Studies of granular materials are intrinsically inter-disciplinary and they borrow ideas and methods from

other fields of physics such as statistical physics, me-chanics, fluid dynamics, and the theory of plasticity. Onthe flip side, progress in understanding granular mat-ter can be often applied to seemingly unrelated phys-ical systems, such as ultra-thin liquid films, foams,colloids, emulsions, suspensions, and other soft con-densed matter systems. The common feature sharedby these systems is the discrete microstructure di-rectly influencing macroscopic behavior. For example,the order parameter description similar to that of Sec.VI.A.1 was applied to stick-slip friction in ultra-thinfilms, (Aranson et al., 2002c; Carlson and Batista, 1996;Israelachvili et al., 1988; Urbach et al., 2004).

Lemaitre (2002); Lemaitre and Carlson (2004) ap-plied the idea of shear-transformation zone (STZ) pio-neered by Falk and Langer (1998) for amorphous solidsboth to granular matter and to the boundary lubrica-tion problem in confined fluid. In this theory the plas-tic deformation is represented by a population of meso-scopic regions which may undergo non-affine deforma-tions in response to stress. Concentration of STZs inamorphous material is somewhat similar to the order pa-rameter (relative concentration of defects) introduced byAranson et al. (2002c). A conceptually similar approachwas proposed by Staron et al. (2002) who described theonset of fluidization as a percolation of the contact net-work with fully mobilized friction. Whereas derivation ofthe constitutive relations from first-principle microscopicrules is still a formidable challenge, these approaches arepromising for understanding of not only the boundarylubrication problem, but also onset of motion in densegranular matter.

Flowing liquid foams and emulsions share many simi-larities with granular matter: they have internal discretestructure (bubbles and drops play the role of grains), andtwo different mechanisms are responsible for the trans-mission of stresses: elastic for small stress and visco-plastic above certain yield stress. However, there areadditional complications: bubbles are highly deformableand, unlike granular matter, a number of particles maychange due to the coalescence of bubbles.

Foams and granular materials often exhibit sim-ilar behavior, such as non-trivial stress relaxationand power-law distribution of rearrangement events(Dennin and Knobler, 1997). Stick-slip behavior was re-ported both for sheared foams (Lauridsen et al., 2002)and granular materials (Nasuno et al., 1997). Remark-ably, recent experiments with two dimensional foams(Lauridsen et al., 2004) and three dimensional emulsions(Coussot et al., 2002a,b; DaCruz , 2002) strongly sug-gest the coexistence between flowing (liquid) and jammed(solid) states reminiscent of that in granular matter.Furthermore, avalanche behavior reminiscent of granularflows down an inclined plane (Daerr and Douady, 1999)was reported by Coussot et al. (2002a) for clay suspen-sions, see Fig. 54. There are many approaches treat-ing foams, gel and suspensions as complex fluids withspecific stress-strain constitutive relation. For example,

34

FIG. 54 Sequence of snapshots illustrating evolution of claysuspension drop poured over sandpaper, from Coussot et al.(2002a)

FIG. 55 A possible phase diagram for jamming. The jammedregion, near the origin, is enclosed by the depicted surface.The line in the temperature-load plane is speculative, andindicates how the yield stress might vary for jammed systemsin which there is thermal motion, from Liu and Nagel (1998)

Fuchs and Cates (2002) used the analogy between glassesand dense colloidal suspensions and applied the modecoupling approach to understand the nonlinear rheologyand yielding. Similar approaches can be possibly usefulfor granular materials (Schofield and Oppenheim, 1994).

Liu and Nagel (1998) suggested that a broad classof athermal soft matter systems (glasses, suspensions,granular materials) shows a universal critical behaviorin the vicinity of solid-fluid or jamming transition, seeFig. 55. Whether jammed systems indeed have com-mon features that can be described by a universal phasediagram is an open issue. An interesting question inthis context is a possibility of thermodynamic descrip-tion of driven, macroscopic, athermal systems like gran-ular materials and foams in terms of some kind of effec-tive temperature. Studies of interacting particles undershear (Corvin et al., 2005; Makse and Kurchan, 2002;O’Hern et al., 2004; Ono et al., 2002; Xu and O’Hern,2005) indicate that indeed under certain conditions it ispossible to define an effective temperature (for example,from the equivalent of the Einstein-Stokes relation) for abroad class of athermal systems from comparison of themechanical linear response with the corresponding time-dependent fluctuation-dissipation relation. However, thepossibility of developing nonequilibrium thermodynamicsof the basis of the effective temperature is under debate.

Granular systems exhibit many similarities with traf-fic flows and collective motion of self-propelled particlessuch as swimming bacteria, fish schools, bird flocks, etc.,see for review (Helbing, 2001). In particular, jammingtransition in granular media and traffic jams show simi-lar features, such as hysteresis, and clusters formation.Moreover, continuum models of traffic flows are oftencast in the form of modified Navier-Stokes equation withdensity-dependent viscosity, similar to granular hydrody-namics.

Let us discuss briefly some open questions in thephysics of granular matter.

• Static vs. dynamic description. Commonly ac-cepted models of rapid granular flows (granular hy-drodynamics) and quasi-static dense flows (elasticand visco-plastic models) are very different, see e.g.Goldenberg and Goldhirsch (2002). However, nearthe fluidization transition, and in dense partially-fluidized flows, the differences between these tworegimes become less obvious. The fluidization of

sheared granular materials has many features of afirst-order phase transition. The phenomenologi-cal partial fluidization theory in principle can bea bridge between the static and dynamic descrip-tions. The order parameter related to the local co-ordination number appears to be one of the hiddenfields required for a consistent description of granu-lar flows. One important question in this regard isthe universality of the fluidization transition in dif-ferent granular systems and geometries. On the op-posite side of the fluidization transition, the staticstate of the granular matter can be described by theorder parameter related to the percentage of staticcontacts with fully activated dry friction (criticalcontacts) (Staron et al., 2002). It was shown thatonce these contacts form a percolation cluster, thegranular pack slips and fluidization occurs. It is ofobvious interest to relate this “static” order param-eter and the “dynamics” order parameter discussedabove. We see one of the main future challenges inthe systematic derivation of the continuum theoryvalid both for flowing and static granular matter.

• Statistical mechanics of dense granular systems.Clearly, discrete grain structure plays a major rolein the dynamics and inherent stochasticity of gran-ular response. The number of particles in a typi-cal granular assembly is large (106 or more) but itis much smaller than the Avogadro number. Tra-ditional tools of statistical physics do not applyto dense granular systems since grains do not ex-hibit thermal Brownian motion. One of the alter-native ways of describing statistics of granular me-dia was suggested by Edwards and Grinev (1998)in which they proposed that volume rather thanenergy serves as the extensive variable in a staticgranular system, so that the role of temperatureis played by the compactivity which is the deriva-tive of the volume with respect to the usual en-tropy. Recent experiments (Makse and Kurchan,2002; Schroter et al., 2005) aim to test this theoryexperimentally. Connecting Edwards theory withgranular hydrodynamics will be an interesting chal-lenge for future studies.

• Realistic simulations of three-dimensional granu-lar flows. Even the most advanced simulations ofgranular flows in three dimensions (Silbert, 2005;Silbert et al., 2003) are limited to relatively smallsamples (e.g. 100 × 40 × 40 particles box) and arevery time consuming. The granular problems areinherently very stiff: while the collisions betweenparticles are very short (O(10−4sec), the collectiveprocesses of interest may take many seconds or min-utes. As a result, to the time step limitations asimulation of realistic hard particles is not feasi-ble: the “simulations” particles have elastic mod-uli several orders of magnitude smaller than sandor glass. The particle softness may introduce un-

35

physical artifacts in the overall picture of the mo-tion. Different approaches to handling this prob-lem will be necessary to advance the state of theart in simulations. New opportunity can be offeredby the equation-free simulation method proposedby Kevrekidis et al. (2004). Another area of sim-ulations which needs further refinements is an ac-curate account of dry friction. In the absence ofa better solution current methods (see for review(Luding, 2004)) employ various approximate tech-niques to simulation dry friction, and accuracy ofthese methods can be questionable.

• Complex interactions. Understanding of dynam-ics of granular systems with complex interactions iscertainly an intriguing and rapidly developing field.While interaction of grains with intersticial fluid isa traditional part of engineering research, effects ofparticle anisotropy, long-range electromagnetic in-teractions mediating collisions, adhesion, agglom-eration and many others constitute a formidablechallenge for theorists and a fertile field of futureresearch.

• Granular physics on a nano-scale. There is apersistent trend in the industry such as powdermetallurgy, pharmaceutical and various chemicaltechnologies towards operating with smaller andsmaller particles. Moreover, it was recognized re-cently that micro- and nano-particles can be use-ful for fabrication of desired ordered structuresand templates for a broad range of nanotech-nological applications through self-assembly pro-cesses. Self-assembly, the spontaneous organiza-tion of materials into complex architectures, con-stitutes one of the greatest hopes of realizing thechallenge to create ever smaller nanostructures.It is a particulary attractive alternative to tra-ditional approaches such as lithography and elec-tron beam writing. Reduction of the particle sizeto micro- and nano -scales shifts the balance be-tween forces controlling particle interaction becausethe dominant interactions depend on the parti-cle size. While for macroscopic grains the dy-namics are governed mostly by the gravity, colli-sional and frictional forces, for micro- and nano-particles the dominant interactions include long-range electromagnetic forces, short- range van derWaals interactions, etc. Nevertheless, some con-cepts and ideas developed in the “traditional”granular physics were successfully applied to un-derstand dynamic self-assembly of microparticles(Sapozhnikov et al., 2003a, 2004) and even biolog-ical microtubules (Aranson and Tsimring, 2005).We expect to see more and more efforts in this di-rection.

Acknowledgments

The authors thank Dmitrii Volfson, Alexey Snezhko,Maksim Saposhnikov, Jie Li, Adrian Daerr, BobBehringer, Jerry Gollub, Thomas Halsey, Denis Ertas,Harry Swinney, Jeff Olafsen, Eli Ben-Naim, Valerii Vi-nokur, Wai Kwok, George Crabtree, Paul Umbanhowar,Francisco Melo, Eric Clement, Jacques Prost, PhilippeClaudin, Julio Ottino, Devang Khakhar, Jean-PhilippeBouchaud, Olivier Pouliquen, Jacques Duran, AnaelLemaitre, Evelyne Kolb, Hugues Chate, Gary Grest, Ar-shad Kudrolli, Douglas Durian, Peter Schiffer, Leo Sil-bert, Wolfgang Losert, Daniel Blair, Paul Chaikin, Hen-rich Jaeger, Sid Nagel, Leo Kadanoff, Thomas Witten,Sue Coppersmith, Baruch Meerson, Ray Goldstein, ChayGoldenberg, Isaak Goldhirsch, Robert Ecke, ThorstenPoschel, Alexandre Valance, James Dufty, James Jenk-ins, Dietrich Wolf, Haye Hinrichsen, Lorenz Kramer, LenPismen, Martin van Hecke, Wim van Saarloos, GuenterAhlers, Jacob Israelachvili, James Langer, Pierre-Gillesde Gennes and many others for useful discussions. Thiswork was supported by the Office of the Basic Energy Sci-ences at the United States Department of Energy, grantsW-31-109-ENG-38, and DE-FG02-04ER46135.

The review was partly written when one of us (I.A) wasattending Granular Session in Institute Henry Poincare,Paris, and Granular Physics Program, Kavli Institute forTheoretical Physics in Santa Barbara.

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